<?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> ; <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">f</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </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="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Olive" title="b1">q</font> );<br/>
<a NAME="E1:46"/>
( <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self" title="HILBERT2:th.34">Th34</a>, <a class="ref" href="trees_9.html#T1" title="TREES_9:th.1">TREES_9:1</a></i>;<br/>
<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ]
 
;<br/>
<a NAME="E3:46"/><i><font color="Green" title="E38">A2</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] holds <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</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">f</font>] implies for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>

<b>given </b><font color="Maroon" title="c4">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> <b> such that </b><a NAME="E2:46_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E3:46_1"/><i><font color="Green" title="E39">A3</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <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"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>

<b>assume </b><a NAME="E4:46_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</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">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<a NAME="E5:46_1"/><i><font color="Green" title="E41">A5</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/>
<a NAME="E6:46_1"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E4:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/>
<a NAME="E7:46_1"/><i><font color="Green" title="E43">A7</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="trees_9.html#T3" title="TREES_9:th.3">TREES_9:3</a></i>;<br/>
<a NAME="E8:46_1"/><i><font color="Green" title="E44">A8</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1"/>
( <font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</a> or <font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</a> or <font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  )
 </span><b>by </b><i><a class="ref" href="hilbert2.html#T9" target="_self" title="HILBERT2:th.9">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/>
<font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V1" title="HILBERT2:attr.1">conjunctive</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>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E3:46_1_1_1"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D6" target="_self" title="HILBERT2:def.6">Def6</a></i>;<br/><a NAME="E4:46_1_1_1"/><i><font color="Green" title="E46">A10</font></i>: 
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c7">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="hilbert2.html#T32" target="_self" title="HILBERT2:th.32">Th32</a></i>;<br/><a NAME="E5:46_1_1_1"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T14" title="TREES_4:th.14">TREES_4:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_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="E1:46_1_1_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="hilbert2.html#T5" target="_self" title="HILBERT2:th.5">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1_1_1_1"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">r</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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Olive" title="b1">r</font> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c6">r</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font> )</span><br/><b>thus </b><font color="Maroon" title="c8">r</font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self" title="HILBERT2:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E8:46_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self" title="HILBERT2:th.7">Th7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font></span><br/><b>thus </b><a NAME="E3:46_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self" title="HILBERT2:th.8">Th8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_1_1_1_2"/><i><font color="Green" title="E48">A13</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">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> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font> )</span><br/><b>thus </b><font color="Maroon" title="c8">s</font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self" title="HILBERT2:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E8:46_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self" title="HILBERT2:th.7">Th7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font></span><br/><b>thus </b><a NAME="E3:46_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self" title="HILBERT2:th.8">Th8</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:46_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</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="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/>
<font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V2" title="HILBERT2:attr.2">conditional</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>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E3:46_1_1_2"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D7" target="_self" title="HILBERT2:def.7">Def7</a></i>;<br/><a NAME="E4:46_1_1_2"/><i><font color="Green" title="E46">A15</font></i>: 
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c7">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="hilbert2.html#T33" target="_self" title="HILBERT2:th.33">Th33</a></i>;<br/><a NAME="E5:46_1_1_2"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T14" title="TREES_4:th.14">TREES_4:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_2_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1_1_2"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="hilbert2.html#T5" target="_self" title="HILBERT2:th.5">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_2_1_1_1"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">r</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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Olive" title="b1">r</font> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c6">r</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font> )</span><br/><b>thus </b><font color="Maroon" title="c8">r</font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self" title="HILBERT2:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E8:46_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self" title="HILBERT2:th.7">Th7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font></span><br/><b>thus </b><a NAME="E3:46_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self" title="HILBERT2:th.8">Th8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2_1_1_2"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">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> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font> )</span><br/><b>thus </b><font color="Maroon" title="c8">s</font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self" title="HILBERT2:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</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="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E8:46_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self" title="HILBERT2:th.7">Th7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font></span><br/><b>thus </b><a NAME="E3:46_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self" title="HILBERT2:th.8">Th8</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:46_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</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="46_1_1_3"><b>suppose </b></a><a NAME="E1:46_1_1_3"/>
<font color="Maroon" title="c4">q</font> is <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</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>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:46_1_1_3"/><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D8" target="_self" title="HILBERT2:def.8">Def8</a></i>;<br/><a NAME="E4:46_1_1_3"/>
 <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_3"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="hilbert2.html#D9" target="_self" title="HILBERT2:def.9">Def9</a></i>;<br/><a NAME="E5:46_1_1_3"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E6:46_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4"><b>suppose </b></a><a NAME="E1:46_1_1_4"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</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>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><div class="add"><a NAME="E2:46_1_1_4"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T30" target="_self" title="HILBERT2:th.30">Th30</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E3:46_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:46"/>
for <font color="Olive" title="b1">f</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S1" target="_self" title="HILBERT2:sch.1">HILBERT2:sch 1</a>(<a class="txt" href="hilbert2.html#E2:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="hilbert2.html#E3:46"><i><font color="Green" title="E38">A2</font></i></a>);<br/></i>
<a NAME="E5:46"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">f</font>]
 
;<br/>
<b>hence </b><a NAME="E6:46"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K3" title="HILBERT2:func.3">Subformulae</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
