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

<span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <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="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b1">X</font> <a href="calcul_1.html#R7" title="CALCUL_1:pred.7">|=</a> <font color="Maroon" title="c1">p</font></span><br/><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c2">X</font> <a href="calcul_1.html#R7" title="CALCUL_1:pred.7">|=</a> <font color="Maroon" title="c1">p</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:50"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="calcul_1.html#R7" title="CALCUL_1:pred.7">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#D10" target="_self" title="CALCUL_1:def.10">Def10</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:50"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c4">PR</font> is  <a href="calcul_1.html#V1" title="CALCUL_1:attr.1">a_proof</a> 
 <span class="kw">and </span><br/><a NAME="E8:50"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="calcul_1.html#D9" target="_self" title="CALCUL_1:def.9">Def9</a></span>;<br/>
<a NAME="E9:50"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="calcul_1.html#D8" target="_self" title="CALCUL_1:def.8">Def8</a></span>;<br/>
<a NAME="E10:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A6</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( $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#K3" title="CQC_LANG:func.3">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="E11:50"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
 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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:50_1"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K3" title="CQC_LANG:func.3">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> )</span><br/>

<a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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#K3" title="CQC_LANG:func.3">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="c6">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> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:50_1"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K3" title="CQC_LANG:func.3">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="c6">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> )</span><br/>

<a NAME="E6:50_1"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
<font color="Maroon" title="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E5:50_1"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E61">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_1_1"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  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#K3" title="CQC_LANG:func.3">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="Maroon" title="c6">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> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  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#K3" title="CQC_LANG:func.3">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="c6">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> ) )</span><br/><span class="kw">thus </span><a NAME="E2:50_1_1"/>
<font color="Maroon" title="c7">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K3" title="CQC_LANG:func.3">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="c6">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> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">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="c6">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#K3" title="CQC_LANG:func.3">CQC-WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="txt" href="calcul_1.html#E49"><span class="lab"><font color="Green" title="E47">Lm3</font></span></a></span>;<br/><a NAME="E4:50_1_1"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c4">PR</font>,<font color="Maroon" title="c6">m</font> <a href="calcul_1.html#R3" title="CALCUL_1:pred.3">is_a_correct_step</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="calcul_1.html#D8" target="_self" title="CALCUL_1:def.8">Def8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_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="c6">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="c6">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="c6">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="c6">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="c6">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="c6">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="c6">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="c6">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="c6">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="c6">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="calcul_1.html#T31" target="_self" title="CALCUL_1:th.31">Th31</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><a NAME="E2:50_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>(  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Olive" title="b1">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="Olive" title="b1">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="c6">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="Olive" title="b1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_1_1_1_1"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#T15" target="_self" title="CALCUL_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><a NAME="E2:50_1_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <font color="Maroon" title="c8">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">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#K5" title="CQC_LANG:func.5">VERUM</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_1_1_1_2"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#T30" target="_self" title="CALCUL_1:th.30">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="c10">f</font>, <font color="Maroon" title="c11">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_3"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_3"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_3"/><span class="lab"><font color="Green" title="E66">A19</font></span>: 
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c10">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="c11">f1</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c10">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="c11">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="c9">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="c10">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="c6">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="c11">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E6:50_1_1_1_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:50_1_1_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_3"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:50_1_1_1_1_3"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_1_1_3"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="calcul_1.html#T16" target="_self" title="CALCUL_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">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="c11">f</font>, <font color="Maroon" title="c12">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E64">A20</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E66">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E67">A23</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E7:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E68">A24</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">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="c12">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="c11">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="c12">f1</font></span>)</span> &amp;  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'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="c11">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="c12">f1</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c11">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="c12">f1</font> &amp; <font color="Maroon" title="c11">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="c10">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c12">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E8:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<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="c11">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="c11">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E9:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E70">A26</font></span>: 
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c11">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">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="c11">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E69">A25</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><a NAME="E10:50_1_1_1_1_4"/><span class="lab"><font color="Green" title="E71">A27</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:50_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E67">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:50_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A28</font></span>: 
 ex <font color="Olive" title="b1">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c10">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="Olive" title="b1">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="Olive" title="b1">h1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E5:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:50_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E10:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:50_1_1_1_1_4"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E68">A24</font></span></a>, <a class="txt" href="#E12:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="txt" href="#E9:50_1_1_1_1_4"><span class="lab"><font color="Green" title="E70">A26</font></span></a>, <a class="ref" href="calcul_1.html#T18" target="_self" title="CALCUL_1:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">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="c11">f</font>, <font color="Maroon" title="c12">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c13">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E64">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E65">A30</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E66">A31</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E67">A32</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E7:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E68">A33</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">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="c11">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="c12">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="c11">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c13">p</font> &amp;  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'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="c11">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="c12">f1</font> &amp; <font color="Maroon" title="c11">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="c10">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c12">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E8:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E69">A34</font></span>: 
<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="c11">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="c13">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E9:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E70">A35</font></span>: 
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c11">f</font></span>)</span> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E69">A34</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><a NAME="E10:50_1_1_1_1_5"/><span class="lab"><font color="Green" title="E71">A36</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:50_1_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E67">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:50_1_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A37</font></span>: 
 ex <font color="Olive" title="b1">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c10">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="Olive" title="b1">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="Olive" title="b1">h1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E5:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:50_1_1_1_1_5"/>
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E64">A29</font></span></a>, <a class="txt" href="#E10:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E71">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:50_1_1_1_1_5"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E68">A33</font></span></a>, <a class="txt" href="#E12:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E72">A37</font></span></a>, <a class="txt" href="#E9:50_1_1_1_1_5"><span class="lab"><font color="Green" title="E70">A35</font></span></a>, <a class="ref" href="calcul_1.html#T19" target="_self" title="CALCUL_1:th.19">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">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="c11">f</font>, <font color="Maroon" title="c12">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E7:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E68">A42</font></span>: 
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">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="c12">f1</font> &amp; <font color="Maroon" title="c11">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="c10">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c12">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E8:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E69">A43</font></span>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c11">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="c11">f</font></span>)</span> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&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="c12">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E9:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c11">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">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="c11">f</font></span>)</span> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&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="c12">f1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><a NAME="E10:50_1_1_1_1_6"/><span class="lab"><font color="Green" title="E71">A45</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:50_1_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:50_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A46</font></span>: 
 ex <font color="Olive" title="b1">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c10">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="Olive" title="b1">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="Olive" title="b1">h1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E5:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:50_1_1_1_1_6"/>
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E10:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:50_1_1_1_1_6"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E12:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="txt" href="#E9:50_1_1_1_1_6"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="calcul_1.html#T20" target="_self" title="CALCUL_1:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="c10">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_7"/><span class="lab"><font color="Green" title="E64">A47</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_7"/><span class="lab"><font color="Green" title="E65">A48</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_7"/><span class="lab"><font color="Green" title="E66">A49</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c12">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="c10">f</font> &amp; <font color="Maroon" title="c10">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_7"/><span class="lab"><font color="Green" title="E67">A50</font></span>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c10">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="c11">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E7:50_1_1_1_1_7"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_7"><span class="lab"><font color="Green" title="E65">A48</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:50_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A51</font></span>: 
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_7"><span class="lab"><font color="Green" title="E64">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:50_1_1_1_1_7"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c10">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_7"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:50_1_1_1_1_7"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A49</font></span></a>, <a class="txt" href="#E8:50_1_1_1_1_7"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="ref" href="calcul_1.html#T21" target="_self" title="CALCUL_1:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="c10">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_8"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_8"/><span class="lab"><font color="Green" title="E65">A53</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_8"/><span class="lab"><font color="Green" title="E66">A54</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c12">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="c10">f</font> &amp; <font color="Maroon" title="c10">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_8"/><span class="lab"><font color="Green" title="E67">A55</font></span>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c10">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="c12">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E7:50_1_1_1_1_8"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_8"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:50_1_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A56</font></span>: 
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_8"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:50_1_1_1_1_8"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c10">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_8"><span class="lab"><font color="Green" title="E67">A55</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:50_1_1_1_1_8"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_1_1_8"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E8:50_1_1_1_1_8"><span class="lab"><font color="Green" title="E68">A56</font></span></a>, <a class="ref" href="calcul_1.html#T22" target="_self" title="CALCUL_1:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="c10">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c11">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_9"/><span class="lab"><font color="Green" title="E64">A57</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_9"/><span class="lab"><font color="Green" title="E65">A58</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_9"/><span class="lab"><font color="Green" title="E66">A59</font></span>: 
(  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">p</font>) &amp; <font color="Maroon" title="c10">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_9"/><span class="lab"><font color="Green" title="E67">A60</font></span>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c10">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="c11">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E7:50_1_1_1_1_9"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_9"><span class="lab"><font color="Green" title="E65">A58</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:50_1_1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A61</font></span>: 
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_9"><span class="lab"><font color="Green" title="E64">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:50_1_1_1_1_9"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c10">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_9"><span class="lab"><font color="Green" title="E67">A60</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:50_1_1_1_1_9"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_1_1_9"><span class="lab"><font color="Green" title="E66">A59</font></span></a>, <a class="txt" href="#E8:50_1_1_1_1_9"><span class="lab"><font color="Green" title="E68">A61</font></span></a>, <a class="ref" href="calcul_1.html#T25" target="_self" title="CALCUL_1:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:50_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="c6">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="c10">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c11">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1_1_10"/><span class="lab"><font color="Green" title="E64">A62</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1_1_10"/><span class="lab"><font color="Green" title="E65">A63</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1_1_10"/><span class="lab"><font color="Green" title="E66">A64</font></span>: 
(  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="substut2.html#K4" title="SUBSTUT2:func.4">.</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>) &amp;  not <font color="Maroon" title="c13">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="c10">f</font></span>)</span> &amp;  not <font color="Maroon" title="c13">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#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">p</font>)</span>)</span> &amp; <font color="Maroon" title="c10">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="c9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1_1_10"/><span class="lab"><font color="Green" title="E67">A65</font></span>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c10">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#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">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="c6">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="calcul_1.html#D7" target="_self" title="CALCUL_1:def.7">Def7</a></span>;<br/><a NAME="E7:50_1_1_1_1_10"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_1_1_10"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:50_1_1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A66</font></span>: 
 ex <font color="Olive" title="b1">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>  st <br/>( <font color="Olive" title="b1">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="c9">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="Olive" title="b1">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="Olive" title="b1">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E3:50_1_1_1_1_10"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:50_1_1_1_1_10"/>
(  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c8">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="c10">f</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_1_1_10"><span class="lab"><font color="Green" title="E67">A65</font></span></a>, <a class="ref" href="calcul_1.html#T5" target="_self" title="CALCUL_1:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:50_1_1_1_1_10"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_1_1_10"><span class="lab"><font color="Green" title="E66">A64</font></span></a>, <a class="txt" href="#E8:50_1_1_1_1_10"><span class="lab"><font color="Green" title="E68">A66</font></span></a>, <a class="ref" href="calcul_1.html#T29" target="_self" title="CALCUL_1:th.29">Th29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:50_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#K3" title="CQC_LANG:func.3">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="c6">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:50_1"/>
( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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#K3" title="CQC_LANG:func.3">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="c6">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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:50_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#K3" title="CQC_LANG:func.3">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="c6">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="txt" href="#E7:50_1"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:50"/><span class="lab"><font color="Green" title="E55">A67</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E13:50"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E54">A7</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E15:50"/><span class="lab"><font color="Green" title="E56">A68</font></span>: 
<font color="Maroon" title="c5">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> 
 <span class="kw">and </span><br/><a NAME="E16:50"/><span class="lab"><font color="Green" title="E57">A69</font></span>: 
 <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c5">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="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E53">A6</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="calcul_1.html#D12" target="_self" title="CALCUL_1:def.12">CALCUL_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">J</font> being    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c6">A</font><br/>  for <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c6">A</font>  st <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c6">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c6">A</font>  st <font color="Maroon" title="c7">J</font>,<font color="Olive" title="b1">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c7">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c6">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )</span><br/>





<span class="kw">assume </span><a NAME="E17:50"/>
<font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E18:50"/><span class="kw">then </span>
 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#K3" title="CQC_LANG:func.3">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#K2" title="RELSET_1:func.2">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="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></span>;<br/>
<a NAME="E19:50"/><span class="kw">then </span>
<font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></span>;<br/>
<a NAME="E20:50"/><span class="kw">then </span>
<font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">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="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E15:50"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:50"/>
<font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c8">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E15:50"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="ref" href="calcul_1.html#D15" target="_self" title="CALCUL_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
