<?xml version="1.0"?>
<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">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">ll</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<br/>







<a NAME="E1:76"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">ll</font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/>
<a NAME="E2:76"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E9:76"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E3:76"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:76"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E10:76"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E5:76"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E6:76"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E7:76"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">ll</font> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:76"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:76"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="substut1.html#T34">SUBSTUT1:34</a></i>;<br/>
<a NAME="E11:76"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/>
<a NAME="E12:76"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:76_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:76_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="sublemma.html#E4:76_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:76_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:76_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:76_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">ll</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:76_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">ll</font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/>
<a NAME="E6:76_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:76_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E10:76"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E8:76_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:76_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>
<a NAME="E10:76_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:76_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T2" target="_self">Th2</a>, <a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:76_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:76"/>
<font color="Maroon">v</font> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ll</font> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
