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

<b>set </b><font color="Maroon">GA</font> =  <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">s</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>   st  <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">s</font> &amp;  <a href="ami_1.html#K6">IC</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5 &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <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="Olive">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Olive">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></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="Olive">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></span>)</span> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>;<br/><b>set </b><font color="Maroon">Cs</font> =  <a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font>;<br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E36</font></i>: 
 <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 ;<br/><b>assume </b><a NAME="E2:21_1"/><i><font color="Green">E37</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5
 ;<br/><b>assume </b><a NAME="E3:21_1"/>
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>assume </b><a NAME="E4:21_1"/>
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><b>assume </b><a NAME="E5:21_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>assume </b><a NAME="E6:21_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>assume </b><a NAME="E7:21_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>
 ;<br/><a NAME="E8:21_1"/><i><font color="Green">E88</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> ,3 <a href="scmpds_2.html#K9">&lt;=0_goto</a> 9
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E89</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span> = 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K9">Following</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K3" target="_self">SBP</a> ,3 <a href="scmpds_2.html#K9">&lt;=0_goto</a> 9</span>)</span>,<font color="Maroon">s</font>
<b>by </b><i/>
;<br/><i><font color="Green">E90</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> = 
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s</font>,9
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 9</span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_6.html#T23">SCMPDS_6:23</a></i>
;<br/><b>take </b><font color="Maroon">n</font> = 1;<br/><b>thus </b> <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 14</span>)</span>
<b>by </b><i>, <a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
 <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
<b>by </b><i>, </i>
;<br/><b>thus </b><a NAME="E12:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i>, <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>;<br/><a NAME="E13:21_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E14:21_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>
<br/>.= 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>
;<br/><b>thus </b><a NAME="E16:21_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:21"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>;<br/><b>set </b><font color="Maroon">yy</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>;<br/><b>set </b><font color="Maroon">Cs</font> =  <a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font>;<br/><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E109</font></i>: 
 <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 ;<br/><b>assume </b><a NAME="E2:21_2_1"/><i><font color="Green">E110</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5
 ;<br/><b>assume </b><a NAME="E3:21_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>assume </b><a NAME="E4:21_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><b>assume </b><a NAME="E5:21_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>assume </b><a NAME="E6:21_2_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>assume </b><a NAME="E7:21_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>
 ;<br/><a NAME="E8:21_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></span>)</span> ) )
 <b>by </b><i>, , , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T21">NAT_1:21</a></i>;<br/><a NAME="E3:21_2_1_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">pn</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:21_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">pn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 ;<br/><a NAME="E6:21_2_1_1_2"/><b>then </b><i><font color="Green">E120</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 7</span>)</span> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8 <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4">goto</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 7</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 7</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 6</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">pn</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 11 )
 <b>by </b><i>, , , , </i>;<br/><b>set </b><font color="Maroon">s8</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8;<br/><i><font color="Green">E121</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> = 
 <a href="scmpds_2.html#K19">ICplusConst</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 7</span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> 5
<b>by </b><i>, </i>
;<br/><a NAME="E8:21_2_1_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8
 <b>by </b><i>, <a class="ref" href="ami_3.html#T38">AMI_3:38</a></i>;<br/><a NAME="E9:21_2_1_1_2"/><i><font color="Green">E130</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>;<br/><a NAME="E10:21_2_1_1_2"/><i><font color="Green">E131</font></i>: 
4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:21_2_1_1_2"/><b>then </b><i><font color="Green">E132</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:21_2_1_1_2"/><i><font color="Green">E133</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>;<br/><b>set </b><font color="Maroon">x1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span>;<br/><b>set </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span>;<br/><i><font color="Green">E134</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
;<br/><i><font color="Green">E135</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
;<br/><a NAME="E15:21_2_1_1_2"/><b>then </b><i><font color="Green">E136</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/><a NAME="E16:21_2_1_1_2"/><b>then </b><i><font color="Green">E137</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E17:21_2_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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="scmp_gcd.html#T4" target="_self">Th4</a>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:21_2_1_1_2"/><i><font color="Green">E138</font></i>: 
(  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></span>)</span> ) )
 <b>by </b><i>, , , <a class="ref" href="scmp_gcd.html#T2" target="_self">Th2</a>, <a class="ref" href="scmp_gcd.html#T3" target="_self">Th3</a>, , , </i>;<br/><b>set </b><font color="Maroon">s9</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span>;<br/><a NAME="E20:21_2_1_1_2"/><i><font color="Green">E139</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><a NAME="E21:21_2_1_1_2"/><i><font color="Green">E140</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><i><font color="Green">E141</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">8 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T6" target="_self">Th6</a></i>
;<br/><a NAME="E23:21_2_1_1_2"/><i><font color="Green">E142</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:21_2_1_1_2"/>
<font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>then </b><i><font color="Green">E143</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, <a class="ref" href="scmp_gcd.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
;<br/><a NAME="E26:21_2_1_1_2"/>
5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 5
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E27:21_2_1_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 5
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E145</font></i>:  <a href="scmpds_3.html#K2">inspos</a> 11 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, <a class="ref" href="scmp_gcd.html#T9" target="_self">Th9</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,<a href="scmpds_1.html#K23">RetIC</a> </span>)</span>
<b>by </b><i>, , , <a class="ref" href="scmpds_1.html#D23">SCMPDS_1:def 23</a></i>
;<br/><i><font color="Green">E146</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,<a href="scmpds_1.html#K23">RetIC</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">11 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T10" target="_self">Th10</a>, </i>
;<br/><b>then </b><i><font color="Green">E147</font></i>:  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> = 
<font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 13</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
<a href="scmp_gcd.html#K3" target="_self">SBP</a> ,2 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> ,6
<b>by </b><i>, </i>
;<br/><i><font color="Green">E148</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">9 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K3" target="_self">SBP</a> ,2 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> ,6</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E149</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<a href="scmpds_1.html#K22">RetSP</a> </span>)</span>
<b>by </b><i>, , , <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_1.html#D22">SCMPDS_1:def 22</a></i>
<br/>.= 
<font color="Maroon">pn</font>
<b>by </b><i>, <a class="ref" href="scmp_gcd.html#T8" target="_self">Th8</a>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
;<br/><a NAME="E33:21_2_1_1_2"/>
 <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p1">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 6</span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E150</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 6</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
;<br/> <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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> = 
 <a href="scmpds_2.html#K18">Next</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T59">SCMPDS_2:59</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">13 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T11" target="_self">Th11</a>, <a class="ref" href="scmpds_4.html#T70">SCMPDS_4:70</a></i>
;<br/><b>then </b><i><font color="Green">E151</font></i>:  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> = 
<font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 14</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
 <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
<b>by </b><i>, </i>
;<br/><div><b>hereby </b>
<div class="add"><b>take </b><font color="Maroon">n</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 10;<br/><b>thus </b><a NAME="E1:21_2_1_1_2_14"/>
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i/>;<br/><a NAME="E2:21_2_1_1_2_14"/><i><font color="Green">E152</font></i>: 
 <a href="scmpds_2.html#K3">DataLoc</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2 <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K1" target="_self">intpos</a> <span class="p1">(<span class="default"><font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:21_2_1_1_2_14"/><b>then </b>
 <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmpds_2.html#K3">DataLoc</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:21_2_1_1_2_14"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i>, , <a class="ref" href="scmpds_2.html#T59">SCMPDS_2:59</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">pn</font>,6</span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T59">SCMPDS_2:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmp_gcd.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T1" target="_self">Th1</a>, , </i>
;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_2_1_1_2_14_2"/><i><font color="Green">E153</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><a NAME="E2:21_2_1_1_2_14_2"/>
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:21_2_1_1_2_14_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <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="E4:21_2_1_1_2_14_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:21_2_1_1_2_14_2"/>
 <a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmp_gcd.html#K3" target="_self">SBP</a> 
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E155</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="scmp_gcd.html#T5" target="_self">Th5</a>, , </i>
;<br/><a NAME="E7:21_2_1_1_2_14_2"/>
<font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pn</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:21_2_1_1_2_14_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmpds_2.html#K3">DataLoc</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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 9</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2
 <b>by </b><i>, , </i>;<br/><a NAME="E9:21_2_1_1_2_14_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 7</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , , , , , </i>;<br/><b>hence </b><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 8</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T66">SCMPDS_2:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="scmpds_2.html#T59">SCMPDS_2:59</a></i>
;<br/><br/></div>
<b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>

<b>assume </b><a NAME="E5:21"/><i><font color="Green">E158</font></i>: 
(  <a href="scmp_gcd.html#K4" target="_self">GCD-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font> &amp;  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2" target="_self">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E7:21"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:21"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  <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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,2</span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span>,3</span>)</span></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K3" target="_self">SBP</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></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">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" target="_self">intpos</a> <font color="Olive">j</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


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