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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1" target="_self">HP-WFF</a>   st <font color="Olive">T</font> is <a href="hilbert1.html#V6" target="_self">Hilbert_theory</a> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:16_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1" target="_self">HP-WFF</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:16_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1" target="_self">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:16_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:16_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1" target="_self">HP-WFF</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1" target="_self">HP-WFF</a>  ;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<b>thus </b><a NAME="E5:16_1_1"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1" target="_self">HP-WFF</a>  holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1" target="_self">HP-WFF</a>   st <font color="Olive">T</font> is <a href="hilbert1.html#V6" target="_self">Hilbert_theory</a> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font> )
 <b>by </b><i/>;<br/>


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