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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E53</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">X</font> iff  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) ) ) &amp; ( 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  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:20_1_2_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) ) &amp; (  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) &amp; ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) ) &amp; (  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font> ) implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:20_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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