<?xml version="1.0"?>
<div class="add">

<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> ,   <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>2</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; <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="Maroon">a<sub>1</sub></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/>
<a NAME="E1:80"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</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">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b2</font>] holds <br/><font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font>
 
;<br/>
<a NAME="E2:80"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:80_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<b>take </b><font color="Maroon">b</font> =  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>;<br/>

<b>thus </b><a NAME="E4:80_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="ref" href="calcul_1.html#T6" target="_self">Th6</a>);<br/></i>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:80"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:80"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 <b>by </b><i><a class="ref" href="finset_1.html#D1">FINSET_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = <font color="Maroon">F1</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">F2</font>;<br/>
<a NAME="E9:80"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T8" target="_self">Th8</a>, <a class="ref" href="calcul_1.html#T9" target="_self">Th9</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<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">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_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="E1:80_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">F2</font></span>)</span>
 ;<br/><a NAME="E2:80_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T8" target="_self">Th8</a>, <a class="ref" href="calcul_1.html#T9" target="_self">Th9</a>, <a class="ref" href="calcul_1.html#T12" target="_self">Th12</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:80_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:80_2"/>
( <font color="Maroon">a</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">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>] )
 <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E7:80_2"/>
<font color="Maroon">b</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#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><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="E1:80_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">b</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> 
 ;<br/><b>consider </b><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="E3:80_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">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="Maroon">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> ) )
 <b>by </b><i><a class="ref" href="calcul_1.html#T12" target="_self">Th12</a></i>;<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> <b> such that </b><br/><a NAME="E5:80_3"/><i><font color="Green">E83</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> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:80_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E7:80_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E8:80_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T7" target="_self">Th7</a>, <a class="ref" href="calcul_1.html#T8" target="_self">Th8</a>, <a class="ref" href="calcul_1.html#T9" target="_self">Th9</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:80"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</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#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E13:80"/><b>then </b><i><font color="Green">E84</font></i>: 
<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>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T10" target="_self">Th10</a>, <a class="ref" href="finset_1.html#D1">FINSET_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4"><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="E1:80_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">b</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> 
 ;<br/><b>consider </b><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="E3:80_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">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="Maroon">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> ) )
 <b>by </b><i><a class="ref" href="calcul_1.html#T13" target="_self">Th13</a></i>;<br/><b>thus </b><a NAME="E4:80_4"/>
<font color="Maroon">b</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:80"/><b>then </b>
 <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>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<b>hence </b><a NAME="E16:80"/>
 <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
