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

<b>reconsider </b><font color="Maroon">nn2</font> = <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">nn2</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>set </b><font color="Maroon">n2</font> =  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b> ex <font color="Olive">q</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> holds <br/><font color="Olive">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) );<br/>
<b>reconsider </b><font color="Maroon">q0</font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E4:16_1_1"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">q0</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D5">AFINSQ_1:def 5</a></i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">q0</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E6:16_1_1"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> holds <br/><font color="Maroon">q0</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q0</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:16_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E8:16_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</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">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

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

<b>then consider </b><font color="Maroon">q2</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:16_1_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">k2</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> holds <br/><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">k2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">k0</font> = <font color="Maroon">k</font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">q2</font> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<i><font color="Green">E43</font></i>:  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q3</font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a></i>
;<br/>
<a NAME="E7:16_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
;<br/>
<a NAME="E8:16_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:16_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E10:16_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E11:16_1_1_1"/>
for <font color="Olive">k2</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> holds <br/><font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">k2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="xcmplx_0.html#K2">+</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">k2</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> )
 ;<br/>

<a NAME="E2:16_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">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">E49</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">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">E50</font></i>: 
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_1"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font>
 <b>by </b><i/>;<br/><a NAME="E8:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E9:16_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:16_1_1_1_2_1_1_1"/>
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , </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">E60</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1_2"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:16_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:16_1_1_1_2_1_1_2"/>
0 <a href="hidden.html#R2">in</a> 1
 <b>by </b><i/>;<br/><a NAME="E7:16_1_1_1_2_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D5">AFINSQ_1:def 5</a></i>;<br/><a NAME="E8:16_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E9:16_1_1_1_2_1_1_2"/>
<font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="afinsq_1.html#D4">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">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1_1"/>
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">q</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:16_1_1"/><i><font color="Green">E63</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">k3</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Olive">k3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn2</font> holds <br/><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k3</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">k3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k3</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">k3</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/>
<a NAME="E12:16_1_1"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:16_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:16_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E15:16_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">nn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn</font> holds <br/><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</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">nn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:16_1_1_2"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn</font> holds <br/><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</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">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn</font>
 ;<br/>

<a NAME="E2:16_1_1_2_1"/>
<font color="Maroon">nn</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:16_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">nn</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:16_1_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1_2_1"/>
<font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


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


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">ss2</font> = <font color="Maroon">q</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">nn2</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E17:16_1_1"/>
 ex <font color="Olive">s</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ex <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font> &amp; <font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">ss2</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E18:16_1_1"/>
 ex <font color="Olive">inn</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">s</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ex <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">a</font> &amp; <font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Olive">inn</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 ;<br/>


</div>
