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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">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> ;<br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="hilbert3.html#NM1" title="HILBERT3:NM.1">SetValuation</a>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="hilbert3.html#M1" title="HILBERT3:mode.1">Permutation</a> of <font color="Maroon" title="c3">V</font>;<br/><b>let </b><font color="Maroon" title="c5">p'</font> be   <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="c3">V</font>,<font color="Maroon" title="c1">p</font>;<br/><b>let </b><font color="Maroon" title="c6">q'</font> be   <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="c3">V</font>,<font color="Maroon" title="c2">q</font>;<br/>











<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c5">p'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c6">q'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">q</font> )
 ;<br/>

<a NAME="E2:46"/>
 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="c3">V</font>,<font color="Maroon" title="c1">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="c3">V</font>,<font color="Maroon" title="c2">q</font> st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K4" title="HILBERT3:func.4">Perm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K4" title="HILBERT3:func.4">Perm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c2">q</font> &amp; <span class="p1">(<span class="default"><a href="hilbert3.html#K4" title="HILBERT3:func.4">Perm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">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; <span class="p1">(<span class="default"><a href="hilbert3.html#K4" title="HILBERT3:func.4">Perm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">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="ref" href="hilbert3.html#D5" target="_self" title="HILBERT3:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:46"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c4">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">p'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c6">q'</font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>


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