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

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E11</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E12</font></i>: 
 <a href="ami_4.html#K1" target="_self">Euclide-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_3.html#D14">AMI_3:def 14</a><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0 ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 ) ) );<br/>
<a NAME="E4:12"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green">E15</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>;<br/>
<a NAME="E1:12_1"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E2:12_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<b>assume </b><a NAME="E4:12_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E5:12_1"/><i><font color="Green">E20</font></i>: 
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0 ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="ami_4.html#E5:12_1"><i><font color="Green">E20</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1"><b>case </b></a><a NAME="E1:12_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E5:12_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ami_4.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:12_1_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E1:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="ami_4.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:12_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E2:12_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ami_4.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:12_1_1_1_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0 ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E3:12_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ami_4.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E6:12_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0 ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 ) ) )
 <b>by </b><i><a class="txt" href="ami_4.html#E1:12_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="ami_4.html#E2:12_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="ami_4.html#E3:12_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ami_4.html#E4:12_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_2"><b>case </b></a><a NAME="E1:12_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E4:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ami_4.html#E5:12_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ami_4.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</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="ami_4.html#E4:12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="ami_4.html#E5:12"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:12"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0 ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 4 ) ) )
 ;<br/>


</div>
