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

<b>let </b><font color="Maroon" title="c13">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">s</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">starts_at</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">s</font> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <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#K11" title="AMI_3:func.11">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="Olive" title="b1">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#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</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>  &amp; <font color="Olive" title="b2">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>  holds <br/><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#K11" title="AMI_3:func.11">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="Olive" title="b1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">y</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E13">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <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#K11" title="AMI_3:func.11">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="Olive" title="b1">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#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</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>  &amp; <font color="Olive" title="b2">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>  holds <br/><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#K11" title="AMI_3:func.11">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="Olive" title="b1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">y</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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#K11" title="AMI_3:func.11">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#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c14">x</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>  &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>  implies <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:13"/><i><font color="Green" title="E15">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#K11" title="AMI_3:func.11">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>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green" title="E16">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#K11" title="AMI_3:func.11">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="E5:13"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c14">x</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> 
 <b>and </b><br/><a NAME="E6:13"/><i><font color="Green" title="E18">A6</font></i>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><br/>

<a NAME="E7:13"/><i><font color="Green" title="E19">A7</font></i>: 
 <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="c15">y</font>
 
<b>by </b><i><a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_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="13_1_1_1"><b>case </b></a><a NAME="E1:13_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><br/><div class="add"><b>hence </b><a NAME="E2:13_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#K11" title="AMI_3:func.11">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> <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#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E15"><i><font color="Green" title="E11">Lm5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><b>case </b></a><a NAME="E1:13_1_1_2"/><i><font color="Green" title="E20">A8</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="card_3.html#K4" title="CARD_3:func.4">K221</a>(the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ) st <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><br/><div class="add"><b>take </b><font color="Maroon" title="c16">s'</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,4; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><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#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E21">A9</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>

<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="ami_4.html#E1:13_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/><a NAME="E4:13_1_1_2"/><i><font color="Green" title="E22">A10</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>;<br/><a NAME="E5:13_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#K11" title="AMI_3:func.11">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#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E3:13_1_1_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/><a NAME="E6:13_1_1_2"/><b>then </b>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <b>by </b><i><a class="txt" href="ami_4.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E4:13_1_1_2"><i><font color="Green" title="E22">A10</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="E7:13_1_1_2"/><b>then </b>
<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#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c16">s'</font>
 <b>by </b><i><a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">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#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#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T87" title="AMI_1:th.87">AMI_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E4:13_1_1_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E10">Lm4</font></i></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#E7:13"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="ami_4.html#E1:13_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="nat_d.html#T32" title="NAT_D:th.32">NAT_D:32</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_3"><b>case </b></a><a NAME="E1:13_1_1_3"/><i><font color="Green" title="E20">A11</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="card_3.html#K4" title="CARD_3:func.4">K221</a>(the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ) st <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><br/><div class="add"><b>take </b><font color="Maroon" title="c16">s'</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,4; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span><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#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E21">A12</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>

<br/>.= 
<font color="Maroon" title="c17">x'</font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:13_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/><a NAME="E4:13_1_1_3"/><i><font color="Green" title="E22">A13</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c16">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>;<br/><a NAME="E5:13_1_1_3"/><b>then </b><i><font color="Green" title="E23">A14</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#K11" title="AMI_3:func.11">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="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#K11" title="AMI_3:func.11">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#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E3:13_1_1_3"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="ami_4.html#E14"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/><a NAME="E6:13_1_1_3"/><b>then </b>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c16">s'</font> <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#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="ami_4.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="ami_4.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="ami_4.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="ami_4.html#E4:13_1_1_3"><i><font color="Green" title="E22">A13</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="E7:13_1_1_3"/><b>then </b><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c16">s'</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">starts_at</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="ami_1.html#D41" title="AMI_1:def.41">AMI_1:def 41</a></i>;<br/><a NAME="E8:13_1_1_3"/><i><font color="Green" title="E25">A16</font></i>: 
 <a href="ami_4.html#K2" title="AMI_4:func.2">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#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="ami_1.html#T86" title="AMI_1:th.86">AMI_1:86</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="E10:13_1_1_3"/><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c16">s'</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</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>,<font color="Maroon" title="c19">k0</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E1:13_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="ami_4.html#E5:13_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="ami_4.html#E7:13_1_1_3"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="ami_4.html#E15"><i><font color="Green" title="E11">Lm5</font></i></a></i>;<br/><a NAME="E11:13_1_1_3"/>
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">k0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c16">s'</font>,<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="E12:13_1_1_3"/><b>then </b><i><font color="Green" title="E27">A18</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_1.html#K7" title="AMI_1:func.7">IC</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 class="p2">(<span class="default"><font color="Maroon" title="c19">k0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E10:13_1_1_3"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="ami_1.html#T91" title="AMI_1:th.91">AMI_1:91</a></i>;<br/><a NAME="E13:13_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#K11" title="AMI_3:func.11">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> <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#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="ami_4.html#E1:13_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="ami_4.html#E5:13_1_1_3"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="ami_4.html#E7:13_1_1_3"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="ami_4.html#E8:13_1_1_3"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="ami_4.html#E15"><i><font color="Green" title="E11">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E14:13_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#K11" title="AMI_3:func.11">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> <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#E12:13_1_1_3"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="ami_1.html#T90" title="AMI_1:th.90">AMI_1:90</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:13"/>
<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#K11" title="AMI_3:func.11">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> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
