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

<b>thus </b><a NAME="E1:16_1_2"/>
for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b3">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="b4">n1</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="b3">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="b3">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="b4">n1</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="b4">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">n1</font> holds <br/><font color="Olive" title="b3">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="b3">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b5">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="b5">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="b5">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">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n1</font> ) &amp;  ex <font color="Olive" title="b3">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="b4">n2</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="b3">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="b3">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="b4">n2</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="b4">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">n2</font> holds <br/><font color="Olive" title="b3">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="b3">s</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Olive" title="b5">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="b5">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="b5">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="b2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n2</font> ) holds <br/><font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c4">w2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:16_1_2_1"/><i><font color="Green" title="E13">A19</font></i>: 
(  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">n1</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">n1</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">n1</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">n1</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="c3">w1</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">n1</font> ) &amp;  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">n2</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">n2</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">n2</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">n2</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="c4">w2</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">n2</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">s1</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> , <font color="Maroon" title="c6">n1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:16_1_2_1"/><i><font color="Green" title="E14">A20</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">s1</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="Maroon" title="c5">s1</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="Maroon" title="c6">n1</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="Maroon" title="c6">n1</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="c6">n1</font> holds <br/><font color="Maroon" title="c5">s1</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="c5">s1</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> ) &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">s2</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> , <font color="Maroon" title="c8">n2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E5:16_1_2_1"/><i><font color="Green" title="E15">A21</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">s2</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="Maroon" title="c7">s2</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="Maroon" title="c8">n2</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="Maroon" title="c8">n2</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="c8">n2</font> holds <br/><font color="Maroon" title="c7">s2</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">s2</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> ) &amp; <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font> )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_2_1"><i><font color="Green" title="E13">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c6">n1</font> 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>, <a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green" title="E14">A20</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> implies <font color="Maroon" title="c5">s1</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> $1 );<br/>
<a NAME="E7:16_1_2_1"/><i><font color="Green" title="E16">A22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/>
<a NAME="E8:16_1_2_1"/><i><font color="Green" title="E17">A23</font></i>: 
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="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_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_2_1_1"/><i><font color="Green" title="E18">A24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2_1_1_1"/><i><font color="Green" title="E19">A25</font></i>: 
<font color="Maroon" title="c10">k</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> <font color="Maroon" title="c9">n</font>
 ;<br/><a NAME="E2:16_1_2_1_1_1"/><i><font color="Green" title="E20">A26</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_2_1_1_1"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_1_1_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c5">s1</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c5">s1</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <font color="Maroon" title="c10">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="Maroon" title="c10">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="Maroon" title="c10">k</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_2_1"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="prgcor_2.html#E2:16_1_2_1_1_1"><i><font color="Green" title="E20">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c5">s1</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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> <font color="Maroon" title="c7">s2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green" title="E15">A21</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_2_1_1"><i><font color="Green" title="E18">A24</font></i></a>, <a class="txt" href="prgcor_2.html#E2:16_1_2_1_1_1"><i><font color="Green" title="E20">A26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c5">s1</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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> <font color="Maroon" title="c7">s2</font> <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_2_1_1_1"><i><font color="Green" title="E19">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="prgcor_2.html#E7:16_1_2_1"><i><font color="Green" title="E16">A22</font></i></a>, <a class="txt" href="prgcor_2.html#E8:16_1_2_1"><i><font color="Green" title="E17">A23</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:16_1_2_1"/>
<font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/>


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


</div>
