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<div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/>





<b>thus </b><a NAME="E1:84"/>
 <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_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="E1:84_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E3:84_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:84_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:84_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<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="E2:84"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:84"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font> &amp; <font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:84"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
