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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1" target="_self">HP-WFF</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="hilbert1.html#V6" target="_self">Hilbert_theory</a>
 ;<br/><a NAME="E2:32_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K5" target="_self">CnPos</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="hilbert1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:32_1"/>
 <a href="hilbert1.html#K5" target="_self">CnPos</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_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="E1:32_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K5" target="_self">CnPos</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:32_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E1:32_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="hilbert1.html#D11" target="_self">Def11</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:32_1"/>
 <a href="hilbert1.html#K5" target="_self">CnPos</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hilbert1.html#E2:32_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:32"/>
(  <a href="hilbert1.html#K5" target="_self">CnPos</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> is <a href="hilbert1.html#V6" target="_self">Hilbert_theory</a> )
 ;<br/>


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