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<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> ,   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $3 is   <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>,$1 &amp; $4 is   <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>,$2 implies  ex <font color="Olive" title="b1">p'</font> being  <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>,$1 ex <font color="Olive" title="b2">q'</font> being  <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>,$2 st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $4 &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> ) ) &amp; ( (  not $3 is   <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>,$1 or  not $4 is   <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>,$2 ) implies $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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> ,   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $3 is   <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>,$1 &amp; $4 is   <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>,$2 implies  ex <font color="Olive" title="b1">p'</font> being  <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>,$1 ex <font color="Olive" title="b2">q'</font> being  <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>,$2 st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $4 &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b2">q'</font> ) ) &amp; ( (  not $3 is   <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>,$1 or  not $4 is   <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>,$2 ) implies $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b5">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]</span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">c</font>]</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_1_1_1_1"/>
( ( <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> &amp; <font color="Maroon" title="c6">b</font> is   <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">q</font> ) or  not <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> or  not <font color="Maroon" title="c6">b</font> is   <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">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c5">a</font> is   <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="c3">p</font>
 <b>and </b><br/><a NAME="E2:39_1_1_1_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c6">b</font> is   <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">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">c</font>]</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c5">a</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="c3">p</font> <b>by </b><i><a class="txt" href="#E1:39_1_1_1_1_1"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c6">b</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">q</font> <b>by </b><i><a class="txt" href="#E2:39_1_1_1_1_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><b>take </b>
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>]</span><br/><b>thus </b><a NAME="E5:39_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/>
(  not <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> or  not <font color="Maroon" title="c6">b</font> is   <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">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">c</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">c</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:39_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b5">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">d</font>]</span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">d</font>]</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_1_1_2_1"/>
( ( <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> &amp; <font color="Maroon" title="c6">b</font> is   <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">q</font> ) or  not <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> or  not <font color="Maroon" title="c6">b</font> is   <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">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_1_2_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">a</font> is   <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="c3">p</font>
 <b>and </b><br/><a NAME="E2:39_1_1_2_1_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c6">b</font> is   <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">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">d</font>]</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c5">a</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="c3">p</font> <b>by </b><i><a class="txt" href="#E1:39_1_1_2_1_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c6">b</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">q</font> <b>by </b><i><a class="txt" href="#E2:39_1_1_2_1_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c8">q'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c8">q'</font>]</span><br/><b>thus </b><a NAME="E5:39_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c8">q'</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2_1_2"/>
(  not <font color="Maroon" title="c5">a</font> is   <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="c3">p</font> or  not <font color="Maroon" title="c6">b</font> is   <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">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">d</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">d</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:39_1_1"/><i><font color="Green" title="E29">A7</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">d</font>] holds <br/><font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font>
 
;<br/>
<a NAME="E4:39_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">d</font>] holds <br/><font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> <b> such that </b><br/><a NAME="E6:39_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> 1
 <b>and </b><br/><a NAME="E7:39_1_1"/><i><font color="Green" title="E32">A10</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" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>and </b><br/><a NAME="E8:39_1_1"/><i><font color="Green" title="E33">A11</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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 <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span>] )
 <b>from </b><i><a class="ref" href="hilbert2.html#S3" title="HILBERT2:sch.3">HILBERT2:sch 3</a>(<a class="txt" href="#E1:39_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E2:39_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E4:39_1_1"><i><font color="Green" title="E30">A8</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font>, <a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> 1 &amp; ( 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" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">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="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font> being  <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="Olive" title="b1">p</font> ex <font color="Olive" title="b4">q'</font> being  <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="Olive" title="b2">q</font> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b4">q'</font> ) ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>3</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><font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E9:39_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E10:39_1_1"/><b>then </b><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="#E6:39_1_1"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>
<a NAME="E11:39_1_1"/><i><font color="Green" title="E35">A13</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>3</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="39_1_1_3"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font>]</span><br/>

<a NAME="E1:39_1_1_3"/>
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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> &amp; <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="txt" href="#E7:39_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1_1_3"/>
<font color="Maroon">S<sub>3</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="ref" href="hilbert3.html#D4" target="_self" title="HILBERT3:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:39_1_1"/><i><font color="Green" title="E36">A14</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>3</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/>( <font color="Maroon">S<sub>3</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>3</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="39_1_1_4"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">s</font>] implies ( <font color="Maroon">S<sub>3</sub></font>[<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>] &amp; <font color="Maroon">S<sub>3</sub></font>[<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><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39_1_1_4"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font>
 <b>and </b><br/><a NAME="E2:39_1_1_4"/><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<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>] &amp; <font color="Maroon">S<sub>3</sub></font>[<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><br/>

<b>consider </b><font color="Maroon" title="c6">p'</font> being   <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>, <font color="Maroon" title="c7">q'</font> being   <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><b> such that </b><br/><a NAME="E4:39_1_1_4"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c7">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E8:39_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="#E1:39_1_1_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E2:39_1_1_4"><i><font color="Green" title="E38">A16</font></i></a></i>;<br/>
<a NAME="E5:39_1_1_4"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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/>
<b>hence </b><a NAME="E6:39_1_1_4"/>
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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>
 <b>by </b><i><a class="txt" href="#E4:39_1_1_4"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="hilbert3.html#T25" target="_self" title="HILBERT3:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<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><br/>

<b>consider </b><font color="Maroon" title="c8">p'</font> being   <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>, <font color="Maroon" title="c9">q'</font> being   <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><b> such that </b><br/><a NAME="E8:39_1_1_4"/><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c8">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c9">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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> <font color="Maroon" title="c8">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c9">q'</font> )
 <b>by </b><i><a class="txt" href="#E8:39_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="#E1:39_1_1_4"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E2:39_1_1_4"><i><font color="Green" title="E38">A16</font></i></a></i>;<br/>
<a NAME="E9:39_1_1_4"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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>hence </b><a NAME="E10:39_1_1_4"/>
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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>
 <b>by </b><i><a class="txt" href="#E8:39_1_1_4"><i><font color="Green" title="E40">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:39_1_1"/><i><font color="Green" title="E37">A19</font></i>: 
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>3</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="#E10:39_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="#E11:39_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="#E12:39_1_1"><i><font color="Green" title="E36">A14</font></i></a>);<br/></i>
<b>thus </b><a NAME="E14:39_1_1"/>
<font color="Maroon" title="c3">M</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font>, <a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> 1 &amp; ( 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" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">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="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font> being  <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="Olive" title="b1">p</font> ex <font color="Olive" title="b4">q'</font> being  <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="Olive" title="b2">q</font> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b4">q'</font> ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>) )</span><br/>

<b>thus </b><a NAME="E1:39_1_1_5"/>
(  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  or <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>) )
 <b>by </b><i><a class="txt" href="#E13:39_1_1"><i><font color="Green" title="E37">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E15:39_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> 1
 <b>by </b><i><a class="txt" href="#E6:39_1_1"><i><font color="Green" title="E31">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">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="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font> being  <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="Olive" title="b1">p</font> ex <font color="Olive" title="b4">q'</font> being  <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="Olive" title="b2">q</font> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b4">q'</font> ) ) )</span><br/>

<b>thus </b><a NAME="E16:39_1_1"/>
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" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">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="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="#E7:39_1_1"><i><font color="Green" title="E32">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   ex <font color="Olive" title="b3">p'</font> being  <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="Olive" title="b1">p</font> ex <font color="Olive" title="b4">q'</font> being  <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="Olive" title="b2">q</font> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b3">p'</font>,<font color="Olive" title="b4">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b4">q'</font> )</span><br/>

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being  <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">p</font> ex <font color="Olive" title="b2">q'</font> being  <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">q</font> st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b1">p'</font>,<font color="Olive" title="b2">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b2">q'</font> )</span><br/>



<a NAME="E17:39_1_1"/><i><font color="Green" title="E38">A20</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" title="HILBERT3:func.2">SetVal</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> )
 
<b>by </b><i><a class="txt" href="#E13:39_1_1"><i><font color="Green" title="E37">A19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">p'</font> being   <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">p</font>, <font color="Maroon" title="c7">q'</font> being   <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">q</font><b> such that </b><br/><a NAME="E19:39_1_1"/><i><font color="Green" title="E39">A21</font></i>: 
( <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN: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">p'</font>,<font color="Maroon" title="c7">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E8:39_1_1"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">p'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q'</font> being  <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">q</font> st <br/>( <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN: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">p'</font>,<font color="Olive" title="b1">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b1">q'</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c7">q'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN: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">p'</font>,<font color="Maroon" title="c7">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> &amp; <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">q'</font> )</span><br/>

<b>thus </b><a NAME="E20:39_1_1"/>
( <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN: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">p'</font>,<font color="Maroon" title="c7">q'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E19:39_1_1"><i><font color="Green" title="E39">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">q'</font></span><br/>

<a NAME="E21:39_1_1"/>
 ex <font color="Olive" title="b1">p'</font> being  <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">p</font> ex <font color="Olive" title="b2">q'</font> being  <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">q</font> st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b2">q'</font> )
 
<b>by </b><i><a class="txt" href="#E8:39_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="#E17:39_1_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:39_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">q'</font>
 <b>by </b><i><a class="txt" href="#E19:39_1_1"><i><font color="Green" title="E39">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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