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

<b>let </b><font color="Maroon">p</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">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:49"/>
<font color="Maroon">p</font> <a href="calcul_1.html#R5" target="_self">is_formal_provable_from</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</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="E3:49"/><i><font color="Green">E52</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="calcul_1.html#R4" target="_self">|-</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">PR</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">PR</font> is <a href="calcul_1.html#V1" target="_self">a_proof</a> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i>, </i>;<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D12" target="_self">CALCUL_1:def 12</a><br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<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>assume </b><a NAME="E6:49"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R6" target="_self">|=</a> <font color="Maroon">X</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">g</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Olive">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Olive">g</font> ) );<br/>
<a NAME="E7:49"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E8:49"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><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>assume </b><a NAME="E1:49_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:49_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</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> <font color="Maroon">PR</font>
 ;<br/>

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

<b>assume </b><a NAME="E3:49_1"/><i><font color="Green">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E4:49_1"/><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E6:49"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:49_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:49_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies  ex <font color="Olive">g</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Olive">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Olive">g</font> ) )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E6:49"><i><font color="Green">E54</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>take </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> ;<br/><b>thus </b><a NAME="E2:49_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 ;<br/><b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="txt" href="calcul_1.html#E7:49"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E4:49_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">PR</font>,<font color="Maroon">m</font> <a href="calcul_1.html#R3" target="_self">is_a_correct_step</a> 
 <b>by </b><i>, <a class="txt" href="calcul_1.html#E7:49"><i><font color="Green">E55</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8 or <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9 )
 </span><b>by </b><i><a class="txt" href="calcul_1.html#E7:49"><i><font color="Green">E55</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</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="E3:49_1_1_1_1_1"/><i><font color="Green">E84</font></i>: 
(  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="calcul_1.html#R1" target="_self">is_tail_of</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><b>thus </b><a NAME="E4:49_1_1_1_1_1"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</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="E3:49_1_1_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> </span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><b>thus </b><a NAME="E4:49_1_1_1_1_2"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">f1</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="E3:49_1_1_1_1_3"/><i><font color="Green">E86</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="calcul_1.html#R2" target="_self">is_Subsequence_of</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f1</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><a NAME="E7:49_1_1_1_1_3"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">f1</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="E3:49_1_1_1_1_4"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; 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#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</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"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_4"/><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_4"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_4"/>
( 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="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then consider </b><font color="Maroon">h1</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="E9:49_1_1_1_1_4"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h1</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_4"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_4"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="calcul_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_5"/><i><font color="Green">E92</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; 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#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f1</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> &amp;  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_5"/><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_5"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_5"/>
( 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="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then consider </b><font color="Maroon">h1</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="E9:49_1_1_1_1_5"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h1</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_5"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_5"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="calcul_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">f1</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="E3:49_1_1_1_1_6"/><i><font color="Green">E96</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; 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#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></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"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_6"/><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_6"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_6"/>
( 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="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then consider </b><font color="Maroon">h1</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="E9:49_1_1_1_1_6"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h1</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_6"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_6"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="calcul_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">q</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="E3:49_1_1_1_1_7"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_7"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_7"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_7"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">q</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="E3:49_1_1_1_1_8"/><i><font color="Green">E102</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></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">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_8"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_8"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_8"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_8"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:49_1_1_1_1_9"/><i><font color="Green">E104</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></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"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_9"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_9"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_9"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_9"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:49_1_1_1_1_10"/><i><font color="Green">E106</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></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="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E4:49_1_1_1_1_10"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">h</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="E6:49_1_1_1_1_10"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">h</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_10"/>
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_10"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, <a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:49_1_1"/>
 ex <font color="Olive">g</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Olive">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Olive">g</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49_1"/>
 ex <font color="Olive">g</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Olive">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Olive">g</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E6:49"><i><font color="Green">E54</font></i></a>, <a class="ref" href="calcul_1.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:49"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="calcul_1.html#D1" target="_self">Def1</a>);<br/></i>
<a NAME="E10:49"/>
<font color="Maroon">PR</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:49"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g</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="E13:49"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E14:49"/>
for <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 <font color="Olive">p</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" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Olive">p</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:49"/><b>then </b>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R6" target="_self">|=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:49"/><b>then </b>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E17:49"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green">E53</font></i></a>, </i>;<br/>


</div>
