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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:49"/>
<font color="Maroon" title="c1">p</font> <a href="calcul_1.html#R5" title="CALCUL_1:pred.5">is_formal_provable_from</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49"/><i><font color="Green" title="E46">A1</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D10" target="_self" title="CALCUL_1:def.10">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">PR</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="calcul_1.html#K4" title="CALCUL_1:func.4">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2" title="CQC_THE1:func.2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c4">PR</font> is <a href="calcul_1.html#V1" title="CALCUL_1:attr.1">a_proof</a> &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="calcul_1.html#D9" target="_self" title="CALCUL_1:def.9">Def9</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D12" target="_self" title="CALCUL_1:def.12">CALCUL_1:def 12</a><br/><b>let </b><font color="Maroon" title="c6">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c5">A</font>;<br/><b>let </b><font color="Maroon" title="c7">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c5">A</font>;<br/>





<b>assume </b><a NAME="E6:49"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c7">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Olive" title="b2">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Olive" title="b2">g</font> ) );<br/>
<a NAME="E7:49"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E8:49"/><i><font color="Green" title="E50">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:49_1"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:49_1"/><i><font color="Green" title="E53">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E4:49_1"/><i><font color="Green" title="E54">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font> )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:49_1"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:49_1"/><b>then </b><i><font color="Green" title="E56">A11</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Olive" title="b1">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Olive" title="b1">g</font> ) )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="E57">A12</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>take </b><font color="Maroon" title="c10">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/><b>thus </b><a NAME="E2:49_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="calcul_1.html#E47"><i><font color="Green" title="E45">Lm1</font></i></a></i>;<br/><a NAME="E4:49_1_1"/><i><font color="Green" title="E58">A13</font></i>: 
<font color="Maroon" title="c4">PR</font>,<font color="Maroon" title="c9">m</font> <a href="calcul_1.html#R3" title="CALCUL_1:pred.3">is_a_correct_step</a> 
 <b>by </b><i><a class="txt" href="calcul_1.html#E5:49"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="calcul_1.html#E4:49_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="calcul_1.html#D8" target="_self" title="CALCUL_1:def.8">Def8</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 )
 </span><b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="calcul_1.html#T31" target="_self" title="CALCUL_1:th.31">Th31</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_1"/><i><font color="Green" title="E59">A14</font></i>: 
(  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c12">f</font> <a href="calcul_1.html#R1" title="CALCUL_1:pred.1">is_tail_of</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E4:49_1_1_1_1_1"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_1"><i><font color="Green" title="E59">A14</font></i></a>, <a class="ref" href="calcul_1.html#T15" target="_self" title="CALCUL_1:th.15">Th15</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_2"/><i><font color="Green" title="E59">A15</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E4:49_1_1_1_1_2"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_2"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="calcul_1.html#T30" target="_self" title="CALCUL_1:th.30">Th30</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c13">f</font>, <font color="Maroon" title="c14">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_3"/><i><font color="Green" title="E59">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font> <a href="calcul_1.html#R2" title="CALCUL_1:pred.2">is_Subsequence_of</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f1</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_3"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_3"/><i><font color="Green" title="E60">A17</font></i>: 
( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c15">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c15">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E7:49_1_1_1_1_3"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_3"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_3"><i><font color="Green" title="E60">A17</font></i></a>, <a class="ref" href="calcul_1.html#T16" target="_self" title="CALCUL_1:th.16">Th16</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c14">f</font>, <font color="Maroon" title="c15">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_4"/><i><font color="Green" title="E59">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c15">f1</font></span>)</span> &amp;  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c15">f1</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c15">f1</font> &amp; <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_4"/><i><font color="Green" title="E60">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_4"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_4"/><i><font color="Green" title="E61">A20</font></i>: 
( <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c16">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c16">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_4"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="calcul_1.html#E4:49_1_1_1_1_4"><i><font color="Green" title="E60">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E9:49_1_1_1_1_4"/><i><font color="Green" title="E62">A21</font></i>: 
( <font color="Maroon" title="c17">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c17">h1</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c17">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_4"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_4"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_4"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_4"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_4"><i><font color="Green" title="E61">A20</font></i></a>, <a class="txt" href="calcul_1.html#E9:49_1_1_1_1_4"><i><font color="Green" title="E62">A21</font></i></a>, <a class="ref" href="calcul_1.html#T18" target="_self" title="CALCUL_1:th.18">Th18</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c14">f</font>, <font color="Maroon" title="c15">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c16">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_5"/><i><font color="Green" title="E59">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c15">f1</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c16">p</font> &amp;  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c15">f1</font> &amp; <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_5"/><i><font color="Green" title="E60">A23</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_5"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_5"/><i><font color="Green" title="E61">A24</font></i>: 
( <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c17">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c17">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_5"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="calcul_1.html#E4:49_1_1_1_1_5"><i><font color="Green" title="E60">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E9:49_1_1_1_1_5"/><i><font color="Green" title="E62">A25</font></i>: 
( <font color="Maroon" title="c18">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c18">h1</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c18">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_5"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_5"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_5"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_5"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_5"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="calcul_1.html#E9:49_1_1_1_1_5"><i><font color="Green" title="E62">A25</font></i></a>, <a class="ref" href="calcul_1.html#T19" target="_self" title="CALCUL_1:th.19">Th19</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c14">f</font>, <font color="Maroon" title="c15">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_6"/><i><font color="Green" title="E59">A26</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c15">f1</font> &amp; <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <span class="p3">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c15">f1</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_6"/><i><font color="Green" title="E60">A27</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_6"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_6"/><i><font color="Green" title="E61">A28</font></i>: 
( <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c16">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c16">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_6"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_6"><i><font color="Green" title="E59">A26</font></i></a>, <a class="txt" href="calcul_1.html#E4:49_1_1_1_1_6"><i><font color="Green" title="E60">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E9:49_1_1_1_1_6"/><i><font color="Green" title="E62">A29</font></i>: 
( <font color="Maroon" title="c17">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c17">h1</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c17">h1</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:49_1_1_1_1_6"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c14">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c15">f1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_6"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E11:49_1_1_1_1_6"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_6"><i><font color="Green" title="E59">A26</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_6"><i><font color="Green" title="E61">A28</font></i></a>, <a class="txt" href="calcul_1.html#E9:49_1_1_1_1_6"><i><font color="Green" title="E62">A29</font></i></a>, <a class="ref" href="calcul_1.html#T20" target="_self" title="CALCUL_1:th.20">Th20</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c13">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c14">p</font>, <font color="Maroon" title="c15">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_7"/><i><font color="Green" title="E59">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c14">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c14">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_7"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_7"/><i><font color="Green" title="E60">A31</font></i>: 
( <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c16">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c16">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_7"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_7"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_7"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_7"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_7"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="calcul_1.html#T21" target="_self" title="CALCUL_1:th.21">Th21</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c13">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c14">p</font>, <font color="Maroon" title="c15">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1_8"/><i><font color="Green" title="E59">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c14">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">q</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_8"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_8"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_8"/><i><font color="Green" title="E60">A33</font></i>: 
( <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c16">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c16">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_8"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_8"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_8"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_8"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_8"><i><font color="Green" title="E60">A33</font></i></a>, <a class="ref" href="calcul_1.html#T22" target="_self" title="CALCUL_1:th.22">Th22</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c13">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c14">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:49_1_1_1_1_9"/><i><font color="Green" title="E59">A34</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c14">p</font> &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_9"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_9"><i><font color="Green" title="E59">A34</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_9"/><i><font color="Green" title="E60">A35</font></i>: 
( <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c17">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c17">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_9"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_9"><i><font color="Green" title="E59">A34</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_9"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_9"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_9"><i><font color="Green" title="E60">A35</font></i></a>, <a class="ref" href="calcul_1.html#T25" target="_self" title="CALCUL_1:th.25">Th25</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" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c13">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c14">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:49_1_1_1_1_10"/><i><font color="Green" title="E59">A36</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> &amp; not <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font></span>)</span> &amp; not <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c14">p</font></span>)</span> &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c14">p</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:49_1_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></i>;<br/><a NAME="E4:49_1_1_1_1_10"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="calcul_1.html#E3:49_1_1_1_1_10"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E6:49_1_1_1_1_10"/><i><font color="Green" title="E60">A37</font></i>: 
( <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c17">h</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c17">h</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:49_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="calcul_1.html#E2:49_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="calcul_1.html#E5:49_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:49_1_1_1_1_10"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c13">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c14">p</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_10"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_1_1_10"/>
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1_1_1_1_10"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1_1_1_1_10"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="calcul_1.html#T29" target="_self" title="CALCUL_1:th.29">Th29</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" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Olive" title="b1">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Olive" title="b1">g</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49_1"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Olive" title="b1">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Olive" title="b1">g</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="txt" href="calcul_1.html#E6:49_1"><i><font color="Green" title="E56">A11</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:49"/><i><font color="Green" title="E51">A38</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="calcul_1.html#E7:49"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="calcul_1.html#E8:49"><i><font color="Green" title="E50">A5</font></i></a>);<br/></i>
<a NAME="E10:49"/>
<font color="Maroon" title="c4">PR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="calcul_1.html#E5:49"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="calcul_1.html#D8" target="_self" title="CALCUL_1:def.8">Def8</a></i>;<br/>
<a NAME="E11:49"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E13:49"/><i><font color="Green" title="E52">A39</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">g</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E9:49"><i><font color="Green" title="E51">A38</font></i></a></i>;<br/>
<a NAME="E14:49"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/><font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c7">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p</font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="calcul_1.html#E6:49"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></i>;<br/>
<a NAME="E15:49"/><b>then </b>
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c7">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></i>;<br/>
<a NAME="E16:49"/><b>then </b>
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c7">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E5:49"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="calcul_1.html#E13:49"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E17:49"/>
<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c7">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:49"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="calcul_1.html#E5:49"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="calcul_1.html#E13:49"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="calcul_1.html#D15" target="_self" title="CALCUL_1:def.15">Def15</a></i>;<br/>


</div>
