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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E1:17_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<a NAME="E2:17_1_1"/><b>then </b>
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="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<b>hence </b><a NAME="E3:17_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  ;<br/>
<a NAME="E3:17_1"/><i><font color="Green">E15</font></i>: 
 <a href="hilbert1.html#K1">HP-WFF</a>  <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="hilbert1.html#D5">HILBERT1:def 5</a></i>;<br/>
<a NAME="E4:17_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:17_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
;<br/>
<b>hence </b><a NAME="E3:17_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:17_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hilbert1.html#V1">with_VERUM</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:17_1_3"/>
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:17"><i><font color="Green">E12</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert2.html#D2" target="_self">HILBERT2:def 2</a><br/>


</div><b>end;</b></div>
<a NAME="E6:17_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hilbert1.html#V2">with_implication</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert2.html#D4" target="_self">HILBERT2:def 4</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:17_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:17_1_4"/><b>then </b><i><font color="Green">E19</font></i>: 
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="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<b>assume </b><a NAME="E3:17_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:17_1_4"/><b>then </b>
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="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<a NAME="E5:17_1_4"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>4</sub></font>]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="hilbert2.html#E2:17_1_4"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hilbert1.html#V3">with_conjunction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert2.html#D5" target="_self">HILBERT2:def 5</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:17_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:17_1_5"/><b>then </b><i><font color="Green">E20</font></i>: 
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="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<b>assume </b><a NAME="E3:17_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:17_1_5"/><b>then </b>
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="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<a NAME="E5:17_1_5"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font>]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E3:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="hilbert2.html#E2:17_1_5"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:17_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="hilbert1.html#V4">with_propositional_variables</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert2.html#D3" target="_self">HILBERT2:def 3</a><br/>

<a NAME="E1:17_1_6"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">c<sub>3</sub></font>]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E2:17"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:17_1_6"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:17_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="hilbert1.html#V5">HP-closed</a>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E4:17_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hilbert2.html#E5:17_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hilbert2.html#E6:17_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="hilbert2.html#E7:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hilbert1.html#D5">HILBERT1:def 5</a></i>;<br/>
<a NAME="E10:17_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="hilbert1.html#K1">HP-WFF</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="hilbert1.html#D6">HILBERT1:def 6</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>

<a NAME="E11:17_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
;<br/>
<a NAME="E12:17_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E10:17_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E13:17_1"/><b>then </b>
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="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<b>hence </b><a NAME="E14:17_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>


</div>
