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

<b>thus </b><a NAME="E1:16_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st ex <font color="Olive">b<sub>3</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>4</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>4</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:16_1_2_1"/><i><font color="Green">E13</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>(  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:16_1_2_1"/><i><font color="Green">E14</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">c<sub>8</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:16_1_2_1"/><i><font color="Green">E15</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_2_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16"><i><font color="Green">E12</font></i></a>, <a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E7:16_1_2_1"/><i><font color="Green">E16</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">E14</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E8:16_1_2_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_2_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></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">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:16_1_2_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:16_1_2_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</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">E14</font></i></a>, <a class="txt" href="prgcor_2.html#E2:16_1_2_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="prgcor_2.html#E1:16_1_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="prgcor_2.html#E2:16_1_2_1_1_1"><i><font color="Green">E20</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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</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">E19</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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prgcor_2.html#E7:16_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="prgcor_2.html#E8:16_1_2_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:16_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:16_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="prgcor_2.html#E5:16_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>


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


</div>
