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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="hilbert3.html#NM1">SetValuation</a>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="hilbert3.html#M1" target="_self">Permutation</a> of <font color="Maroon">V</font>;<br/>







<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:70"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">q</font>
 ;<br/>

<b>assume </b><a NAME="E3:70"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> is <a href="hilbert3.html#V2" target="_self">pseudo-canonical</a>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:70"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E6:70"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T9" target="_self">Th9</a>, <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/>
<a NAME="E7:70"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:70"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E10:70"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T7" target="_self">Th7</a>, <a class="ref" href="hilbert3.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E11:70"/>
contradiction
 <b>by </b><i><a class="ref" href="hilbert3.html#T8" target="_self">Th8</a></i>;<br/>


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