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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> 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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</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>  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> ) );<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span> ;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">f</font>
 ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">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="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:79_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="calcul_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E5:79_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="calcul_1.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:79_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Y</font>]
 <b>by </b><i><a class="ref" href="calcul_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E5:79_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:79"/>
 <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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