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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/>

<a NAME="E2:41"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="calcul_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>;<br/>
<a NAME="E4:41"/><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T10">CALCUL_1:10</a></i>;<br/>
<a NAME="E5:41"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_2.html#K5" target="_self">Impl</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p0">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="calcul_2.html#E3:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="calcul_2.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E6:41"/><i><font color="Green">E35</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_2.html#E4:41"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E8:41"/><i><font color="Green">E36</font></i>: 
(  <a href="calcul_2.html#K5" target="_self">Impl</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p5">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="calcul_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> ) );<br/>
<a NAME="E9:41"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E10:41"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:41_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 ;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:41_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E6:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E5:41_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:41_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><b>then </b><font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>
<br/>.= 
 <a href="calcul_2.html#K5" target="_self">Impl</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a></i>
;<br/><b>hence </b><a NAME="E7:41_1_1"/>
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E5:41"><i><font color="Green">E34</font></i></a>, <a class="txt" href="calcul_2.html#E1:41_1_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:41_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E2:41_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></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="E5:41_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E1:41_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E6:41_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:41_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:41_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><a NAME="E8:41_1_2"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:41_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E9:41_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="calcul_2.html#E9:41_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E11:41_1_2"/><i><font color="Green">E47</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E9:41_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:41_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E2:41_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E13:41_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></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="E15:41_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="txt" href="calcul_2.html#E11:41_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E16:41_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E5:41_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="calcul_2.html#E7:41_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="calcul_2.html#E15:41_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>;<br/><a NAME="E17:41_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="txt" href="calcul_2.html#E11:41_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="calcul_2.html#E13:41_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><b>then </b><i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p0">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="calcul_2.html#E15:41_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p5">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

;<br/><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p5">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>

;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><a NAME="E23:41_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E9:41_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E24:41_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p5">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E25:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E26:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E27:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E28:41_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:41_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="calcul_2.html#E21:41_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E29:41_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E30:41_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E20:41_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/><a NAME="E31:41_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E28:41_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/> <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="calcul_2.html#E1:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="calcul_2.html#T31" target="_self">Th31</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E33:41_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E31:41_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E35:41_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E30:41_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E36:41_1_2"/>
 <a href="calcul_1.html#K2">Suc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E37:41_1_2"/><b>then </b>
 <a href="calcul_1.html#K2">Suc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="calcul_1.html#R1">is_tail_of</a>  <a href="calcul_1.html#K1">Ant</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E35:41_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="calcul_1.html#D3">CALCUL_1:def 3</a></i>;<br/><a NAME="E38:41_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T33">CALCUL_1:33</a></i>;<br/><a NAME="E39:41_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E5:41_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="calcul_2.html#E16:41_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="calcul_1.html#T56">CALCUL_1:56</a></i>;<br/><a NAME="E40:41_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E41:41_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E2:41_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E42:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><a NAME="E43:41_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E15:41_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="calcul_2.html#E40:41_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><b>thus </b><a NAME="E44:41_1_2"/>
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E39:41_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="calcul_2.html#E43:41_1_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41_1"/>
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:41_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:41"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="calcul_2.html#E9:41"><i><font color="Green">E37</font></i></a>, <a class="txt" href="calcul_2.html#E10:41"><i><font color="Green">E38</font></i></a>);<br/></i>
<a NAME="E12:41"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_2.html#E6:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></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="E14:41"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E11:41"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E15:41"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_2.html#E14:41"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="calcul_2.html#K4" target="_self">Begin</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_2.html#E6:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="calcul_2.html#E8:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>;<br/>
<a NAME="E17:41"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="calcul_2.html#E6:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="calcul_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E18:41"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>;<br/>
<a NAME="E19:41"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E20:41"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K6" target="_self">IdFinS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E21:41"/>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="calcul_2.html#E14:41"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>


</div>
