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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:60"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K6">HP_TAUT</a> 
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Olive">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> </span>}</span> ;<br/>
<a NAME="E2:60"/>
<span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Olive">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Olive">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> </span>}</span> 
 ;<br/>

<a NAME="E2:60_1"/><b>then </b>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> )
 
;<br/>
<b>hence </b><a NAME="E3:60_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Olive">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  ;<br/>
<a NAME="E4:60"/>
<font color="Maroon">X</font> is <a href="hilbert1.html#V6">Hilbert_theory</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_2"/>
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert1.html#D10">HILBERT1:def 10</a><br/>

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>





<a NAME="E2:60_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><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> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><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> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E6:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E8:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E10:60_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span></span>)</span> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:60_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E12:60_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E13:60_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">s</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> )
 
;<br/>
<b>assume </b><a NAME="E14:60_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E15:60_2"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> &amp; <font color="Olive">s</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> )
 
;<br/>
<a NAME="E16:60_2"/><b>then </b>
<font color="Maroon">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E17:60_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:60"/><b>then </b>
 <a href="hilbert1.html#K6">HP_TAUT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="hilbert1.html#T13">HILBERT1:13</a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:60"/><b>then </b>
 ex <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="hilbert3.html#V1" target="_self">canonical</a> )
 
;<br/>
<b>hence </b><a NAME="E8:60"/>
<font color="Maroon">p</font> is <a href="hilbert3.html#V1" target="_self">canonical</a>
 ;<br/>


</div>
