<?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="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">starts_at</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E13">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="E3:12"/><i><font color="Green" title="E14">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#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp; <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#K12" title="AMI_3:func.12">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="E4:12"/><i><font color="Green" title="E15">A4</font></i>: 
( <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> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c17">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#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E16">A5</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">y'</font>
 
<b>by </b><i><a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c17">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">x'</font>
 
<b>by </b><i><a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> $1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> $1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span></span>)</span>;<br/>
<i><font color="Green" title="E18">A7</font></i>: <font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c16">x'</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<i><font color="Green" title="E19">A8</font></i>: <font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c17">y'</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<div><i><font color="Green" title="E20">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">k</font> be    <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> ;<br/><a NAME="E1:12_3"/><i><font color="Green" title="E21">A10</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a></i>;<br/><b>assume </b><a NAME="E2:12_3"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E3:12_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>)
 <b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E1:12_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E10">Lm4</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/><a NAME="E4:12_3"/><i><font color="Green" title="E23">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a></i>;<br/><a NAME="E5:12_3"/><i><font color="Green" title="E24">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E4:12_3"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></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#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E1:12_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="ami_4.html#E2:12_3"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E10">Lm4</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>)
<b>by </b><i><a class="txt" href="ami_4.html#E1:12_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="ami_4.html#E5:12_3"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="int_2.html#T49" title="INT_2:th.49">INT_2:49</a></i>
;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c18">k</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="E12:12"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x'</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">y'</font>
 <b>and </b><br/><a NAME="E13:12"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>from </b><i><a class="ref" href="ami_4.html#S1" target="_self" title="AMI_4:sch.1">AMI_4:sch 1</a>(<a class="txt" href="ami_4.html#E6:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="ami_4.html#E7:12"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="ami_4.html#E8:12"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="ami_4.html#E9:12"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="ami_4.html#E10:12"><i><font color="Green" title="E20">A9</font></i></a>);<br/></i>
<a NAME="E14:12"/><i><font color="Green" title="E23">A16</font></i>: 
(  <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="c16">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="c17">y'</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E15:12"/><i><font color="Green" title="E24">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a></i>;<br/>
<a NAME="E16:12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="ami_4.html#E13:12"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/>
<a NAME="E17:12"/><b>then </b><i><font color="Green" title="E25">A18</font></i>: 
 <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#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> 4
 
<b>by </b><i><a class="txt" href="ami_4.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="ami_4.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a></i>;<br/>
<a NAME="E18:12"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> 4
 
<b>by </b><i><a class="txt" href="ami_4.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ami_4.html#E7"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">k</font></span>)</span></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#K12" title="AMI_3:func.12">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E17:12"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="ami_1.html#T87" title="AMI_1:th.87">AMI_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c16">x'</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">y'</font>
<b>by </b><i><a class="txt" href="ami_4.html#E12:12"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="ami_4.html#E15:12"><i><font color="Green" title="E24">A17</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="txt" href="ami_4.html#E14:12"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>
;<br/><br/>

<b>thus </b><a NAME="E20:12"/>
 ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c13">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">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#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E17:12"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="ami_4.html#E18:12"><i><font color="Green" title="E26">A19</font></i></a></i>;<br/>


</div>
