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

<b>reconsider </b><font color="Maroon" title="c3">nn2</font> = <font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">nn</font> = <font color="Maroon" title="c3">nn2</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">n2</font> =  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b> ex <font color="Olive" title="b1">q</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#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Olive" title="b1">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> holds <br/><font color="Olive" title="b1">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> ) );<br/>
<b>reconsider </b><font color="Maroon" title="c6">q0</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> as   <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>  ;<br/>
<a NAME="E4:16_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">q0</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">q0</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E6:16_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> holds <br/><font color="Maroon" title="c6">q0</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q0</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:16_1_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E5:16_1_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<a NAME="E8:16_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">q2</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="E3:16_1_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> holds <br/><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">k2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">k0</font> = <font color="Maroon" title="c7">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">q3</font> = <font color="Maroon" title="c8">q2</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> as   <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>  ;<br/>
<i><font color="Green" title="E17">A6</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q3</font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a></i>
;<br/>
<a NAME="E7:16_1_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E8:16_1_1_1"/><b>then </b>
0 <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="c8">q2</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="prgcor_2.html#T1" target="_self" title="PRGCOR_2:th.1">Th1</a></i>;<br/>
<a NAME="E9:16_1_1_1"/><b>then </b>
0 <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="c8">q2</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E10:16_1_1_1"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E11:16_1_1_1"/>
for <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> holds <br/><font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">k2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_1_2"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> )
 ;<br/>

<a NAME="E2:16_1_1_1_2"/><b>then </b><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_2_1_1"/>
( <font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c11">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 </span><b>by </b><i><a class="txt" href="prgcor_2.html#E2:16_1_1_1_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_1_2_1_1_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="c8">q2</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="prgcor_2.html#T1" target="_self" title="PRGCOR_2:th.1">Th1</a></i>;<br/><a NAME="E4:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="c8">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:16_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">k2</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="c8">q2</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k2</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="c8">q2</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" target="_self" title="PRGCOR_2:th.1">Th1</a></i>;<br/><a NAME="E8:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k2</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="c8">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E9:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c11">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c11">k2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:16_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c11">k2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_1_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_1_1_2_1_1_1"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_1_2_1_1_2"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c11">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_2"/><i><font color="Green" title="E22">A13</font></i>: 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <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="E4:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">k2</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="c8">q2</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="prgcor_2.html#T1" target="_self" title="PRGCOR_2:th.1">Th1</a></i>;<br/><a NAME="E5:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c11">k2</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="c8">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_2"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" target="_self" title="PRGCOR_2:th.1">Th1</a></i>;<br/><a NAME="E7:16_1_1_1_2_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>;<br/><a NAME="E8:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="prgcor_2.html#E2:16_1_1_1_2_1_1_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E9:16_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c11">k2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_1_1_2_1_1_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1_1_1_2"/>
<font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q3</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c11">k2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:16_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="prgcor_2.html#E6:16_1_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="prgcor_2.html#E10:16_1_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1_1"/>
for <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="prgcor_2.html#E7:16_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="prgcor_2.html#E8:16_1_1"><i><font color="Green" title="E15">A4</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c7">q</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="E11:16_1_1"/><i><font color="Green" title="E16">A15</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Olive" title="b1">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nn2</font> holds <br/><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k3</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">k3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k3</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k3</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/>
<a NAME="E12:16_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E13:16_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:16_1_1"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E15:16_1_1"/><i><font color="Green" title="E18">A17</font></i>: 
( <font color="Maroon" title="c4">nn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">nn</font> holds <br/><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_2"/>
<font color="Maroon" title="c4">nn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:16_1_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">nn</font> holds <br/><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_2_1"/><i><font color="Green" title="E19">A18</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">nn</font>
 ;<br/>

<a NAME="E2:16_1_1_2_1"/>
<font color="Maroon" title="c4">nn</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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">a</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:16_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nn</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:16_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E14:16_1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_2_1"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1_2_1"/>
<font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E11:16_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_1_2_1"><i><font color="Green" title="E19">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c8">ss2</font> = <font color="Maroon" title="c7">q</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c3">nn2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E17:16_1_1"/>
 ex <font color="Olive" title="b1">s</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>  ex <font color="Olive" title="b2">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">s</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">a</font> &amp; <font color="Olive" title="b1">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 &amp; ( <font color="Olive" title="b2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c8">ss2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E15:16_1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="prgcor_2.html#E11:16_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="prgcor_2.html#E14:16_1_1"><i><font color="Green" title="E17">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:16_1_1"/>
 ex <font color="Olive" title="b1">inn</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">s</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>  ex <font color="Olive" title="b3">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">s</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">a</font> &amp; <font color="Olive" title="b2">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> 0 &amp; ( <font color="Olive" title="b3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> holds <br/><font color="Olive" title="b2">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Olive" title="b1">inn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> )
 ;<br/>


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