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

<b>let </b><font color="Maroon" title="c1">p</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>assume </b><a NAME="E1:60"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K6" title="HILBERT1:func.6">HP_TAUT</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c2">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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>  : <font color="Olive" title="b1">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> </span>}</span> ;<br/>
<a NAME="E2:60"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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>  : <font color="Olive" title="b1">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="hilbert1.html#K1" title="HILBERT1:func.1">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" title="c3">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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>  : <font color="Olive" title="b1">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> </span>}</span> 
 ;<br/>

<a NAME="E2:60_1"/><b>then </b>
 ex <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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">p</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> )
 
;<br/>
<b>hence </b><a NAME="E3:60_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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>  : <font color="Olive" title="b1">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  ;<br/>
<a NAME="E4:60"/>
<font color="Maroon" title="c3">X</font> is <a href="hilbert1.html#V6" title="HILBERT1:attr.6">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" title="HILBERT1:func.2">VERUM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert1.html#D10" title="HILBERT1:def.10">HILBERT1:def 10</a><br/>

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ;<br/>





<a NAME="E2:60_2"/>
<font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">p</font></span>)</span> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T41" target="_self" title="HILBERT3:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E4:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<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="c6">r</font></span>)</span></span>)</span> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T42" target="_self" title="HILBERT3:th.42">Th42</a></i>;<br/>
<b>hence </b><a NAME="E5:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<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="c6">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E6:60_2"/>
<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">p</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T43" target="_self" title="HILBERT3:th.43">Th43</a></i>;<br/>
<b>hence </b><a NAME="E7:60_2"/>
<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E8:60_2"/>
<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T44" target="_self" title="HILBERT3:th.44">Th44</a></i>;<br/>
<b>hence </b><a NAME="E9:60_2"/>
<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="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E10:60_2"/>
<font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<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></span>)</span> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T45" target="_self" title="HILBERT3:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E11:60_2"/>
<font color="Maroon" title="c4">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>assume </b><a NAME="E12:60_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E13:60_2"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">s</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> )
 
;<br/>
<b>assume </b><a NAME="E14:60_2"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E15:60_2"/><b>then </b>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Olive" title="b1">s</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> )
 
;<br/>
<a NAME="E16:60_2"/><b>then </b>
<font color="Maroon" title="c5">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E13:60_2"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="hilbert3.html#T46" target="_self" title="HILBERT3:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E17:60_2"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:60"/><b>then </b>
 <a href="hilbert1.html#K6" title="HILBERT1:func.6">HP_TAUT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="hilbert1.html#T13" title="HILBERT1:th.13">HILBERT1:13</a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:60"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a> )
 
;<br/>
<b>hence </b><a NAME="E8:60"/>
<font color="Maroon" title="c1">p</font> is <a href="hilbert3.html#V1" title="HILBERT3:attr.1">canonical</a>
 ;<br/>


</div>
