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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E1:46"/>
( <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self">Th34</a>, <a class="ref" href="trees_9.html#T1">TREES_9:1</a></i>;<br/>
<a NAME="E2:46"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> ]
 
;<br/>
<a NAME="E3:46"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K12">*&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">c<sub>3</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><a NAME="E1:46_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:46_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:46_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:46_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E2:46_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<a NAME="E5:46_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E6:46_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E2:46_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="hilbert2.html#E3:46_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="trees_9.html#T3">TREES_9:3</a></i>;<br/>
<a NAME="E7:46_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E2:46_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="trees_2.html#D11">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">c<sub>4</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 </span><b>by </b><i><a class="ref" href="hilbert2.html#T9" target="_self">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">c<sub>4</sub></font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:46_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E4:46_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="trees_4.html#K10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hilbert2.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:46_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K15">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T14">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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hilbert2.html#T5" target="_self">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">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>
;<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" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<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">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>
;<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" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<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">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></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">c<sub>4</sub></font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:46_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E4:46_1_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="trees_4.html#K10">-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hilbert2.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E5:46_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K15">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T14">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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hilbert2.html#E5:46_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hilbert2.html#T5" target="_self">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">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>
;<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" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<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">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="hilbert2.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert2.html#E7:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>
;<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" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:46_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert2.html#E4:46_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbert2.html#E1:46_1_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<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">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></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">c<sub>4</sub></font> is <a href="hilbert2.html#V3" target="_self">simple</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:46_1_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="hilbert2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E4:46_1_1_3"/>
 <a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:46_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hilbert2.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E5:46_1_1_3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">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">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<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">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_1_4"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T30" target="_self">Th30</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">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">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E5:46_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:46"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S1" target="_self">HILBERT2:sch 1</a>(<a class="txt" href="hilbert2.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert2.html#E3:46"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E5:46"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 
;<br/>
<b>hence </b><a NAME="E6:46"/>
<span class="p1">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K3" target="_self">Subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K3" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
