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

<b>let </b><font color="Maroon" title="c1">Fr</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">absFr</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">absFr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c1">Fr</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Olive" title="b1">absFr</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">absFr</font> holds <br/><font color="Olive" title="b1">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E1:70"/><i><font color="Green" title="E58">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c2">absFr</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:70"/><i><font color="Green" title="E59">A2</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font>
 <b>and </b><br/><a NAME="E4:70"/><i><font color="Green" title="E60">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="#E1:70"><i><font color="Green" title="E58">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">absFr</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c1">Fr</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c2">absFr</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font> holds <br/><font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> $1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> $1</span>)</span> );<br/>
<a NAME="E5:70"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1"/>
(  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="catalan2.html#E56"><i><font color="Green" title="E48">Lm6</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E2:70_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:70"/><i><font color="Green" title="E62">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:70_2"/><i><font color="Green" title="E63">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c4">i1</font> = <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>assume </b><a NAME="E2:70_2"/>
<font color="Maroon" title="c3">i</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="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E3:70_2"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:70_2"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E5:70_2"/><b>then </b>
(  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> &amp;  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:70"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E4:70"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E1:70_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="catalan2.html#E58"><i><font color="Green" title="E50">Lm8</font></i></a></i>;<br/>
<a NAME="E6:70_2"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T142" title="COMPLEX1:th.142">COMPLEX1:142</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:70_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:70"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:70"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E6:70"><i><font color="Green" title="E62">A5</font></i></a>);<br/></i>
<a NAME="E8:70"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">absFr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> &amp; <font color="Maroon" title="c1">Fr</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Fr</font> &amp; <font color="Maroon" title="c2">absFr</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">absFr</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E9:70"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">Fr</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c1">Fr</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c2">absFr</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">absFr</font> holds <br/><font color="Maroon" title="c2">absFr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E3:70"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E4:70"><i><font color="Green" title="E60">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
