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

<b>let </b><font color="Maroon" title="c13">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#R1" title="AMI_3:pred.1">starts_at</a>  <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> 0
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E15">A2</font></i>: 
 <a href="ami_4.html#K1" title="AMI_4:func.1">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">s</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:15"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">x</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>case </b></a><a NAME="E1:15_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>case </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c16">s'</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 4;<br/><b>reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c15">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E3:15_1_1_2"/>
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x'</font> &amp;  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y'</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E21">A8</font></i>: <font color="Maroon" title="c14">x</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c15">y</font> = 
<font color="Maroon" title="c17">x'</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c18">y'</font>
<b>by </b><i><a class="ref" href="ami_4.html#T2" target="_self" title="AMI_4:th.2">Th2</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E1:15_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/><a NAME="E5:15_1_1_2"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> &amp; <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>;<br/><a NAME="E6:15_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">s'</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E4:15_1_1_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E11">Lm3</font></i></a></i>;<br/><a NAME="E7:15_1_1_2"/><b>then </b>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> 4
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E5:15_1_1_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="ami_4.html#E12"><i><font color="Green" title="E10">Lm2</font></i></a></i>;<br/><a NAME="E8:15_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#R2" title="AMI_3:pred.2">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c16">s'</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> = 
<font color="Maroon" title="c16">s'</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_3.html#T44" title="AMI_3:th.44">AMI_3:44</a></i>
<br/>.= 
<font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E5:15_1_1_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E11">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c15">y</font></span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E1:15_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_3"><b>case </b></a><a NAME="E1:15_1_1_3"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">x</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c16">s'</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 4;<br/><b>reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c15">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E3:15_1_1_3"/>
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x'</font> &amp;  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y'</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E21">A11</font></i>: <font color="Maroon" title="c14">x</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c15">y</font> = 
<font color="Maroon" title="c17">x'</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c18">y'</font>
<b>by </b><i><a class="ref" href="ami_4.html#T2" target="_self" title="AMI_4:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c17">x'</font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:15_1_1_3"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/><a NAME="E5:15_1_1_3"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> &amp; <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>;<br/><a NAME="E6:15_1_1_3"/><b>then </b><i><font color="Green" title="E23">A13</font></i>: 
( <font color="Maroon" title="c16">s'</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c16">s'</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E4:15_1_1_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E11">Lm3</font></i></a></i>;<br/><a NAME="E7:15_1_1_3"/><b>then </b>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> 0
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:15"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:15"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ami_4.html#E5:15_1_1_3"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="ami_4.html#E12"><i><font color="Green" title="E10">Lm2</font></i></a></i>;<br/><a NAME="E8:15_1_1_3"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c16">s'</font> <a href="ami_3.html#R1" title="AMI_3:pred.1">starts_at</a>  <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> 0
 <b>by </b><i><a class="ref" href="ami_3.html#D14" title="AMI_3:def.14">AMI_3:def 14</a></i>;<br/><a NAME="E9:15_1_1_3"/><i><font color="Green" title="E25">A15</font></i>: 
 <a href="ami_4.html#K1" title="AMI_4:func.1">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">s'</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="ami_3.html#T43" title="AMI_3:th.43">AMI_3:43</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">k0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:15_1_1_3"/><i><font color="Green" title="E26">A16</font></i>: 
<font color="Maroon" title="c16">s'</font> <a href="ami_3.html#R2" title="AMI_3:pred.2">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c16">s'</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c19">k0</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E1:15_1_1_3"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="ami_4.html#E6:15_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="ami_4.html#E8:15_1_1_3"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><a NAME="E12:15_1_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c16">s'</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c19">k0</font>
 <b>by </b><i><a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><a NAME="E13:15_1_1_3"/><b>then </b><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#R2" title="AMI_3:pred.2">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E11:15_1_1_3"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="ami_3.html#T48" title="AMI_3:th.48">AMI_3:48</a></i>;<br/><a NAME="E14:15_1_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c16">s'</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ami_4.html#E1:15_1_1_3"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="ami_4.html#E6:15_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="ami_4.html#E8:15_1_1_3"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="ami_4.html#E9:15_1_1_3"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E15:15_1_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E13:15_1_1_3"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="ami_3.html#T47" title="AMI_3:th.47">AMI_3:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:15"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
 ;<br/>


</div>
