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<span class="kw">let </span><font color="Maroon" title="c13">s</font> be   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st  <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="Olive" title="b1">P</font> holds <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">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="b2">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="b3">y</font> &amp; <font color="Olive" title="b2">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="b3">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="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Olive" title="b1">P</font>,<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="b2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b3">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c14">P</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="c14">P</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="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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/>



<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <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="c14">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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/>

<span class="kw">let </span><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c15">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="c16">y</font> &amp; <font color="Maroon" title="c15">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="c16">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="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( <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="c15">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="c16">y</font> )
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c15">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> 
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c16">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/>

<a NAME="E5:12"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">y</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> or <font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">x</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><span class="kw">case </span></a><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_1_1_1"/>
<span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="ami_4.html#E13"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><span class="kw">case </span></a><a NAME="E1:12_1_1_2"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x9</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y9</font> = <font color="Maroon" title="c16">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">s9</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,4); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/><a NAME="E3:12_1_1_2"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T2" title="EXTPRO_1:th.2">EXTPRO_1:2</a></span>;<br/><a NAME="E4:12_1_1_2"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c19">s9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<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>)
 ;<br/><font color="Maroon" title="c15">x</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c17">x9</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c18">y9</font>

<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>
;<br/><a NAME="E6:12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">s9</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:12_1_1_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:12_1_1_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="ami_4.html#E12"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E7:12_1_1_2"/><span class="kw">then </span>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c19">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:12_1_1_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="ami_4.html#E11"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E8:12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">P</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c19">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="ami_4.html#E5"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c19">s9</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T18" title="EXTPRO_1:th.18">EXTPRO_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:12_1_1_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:12_1_1_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="ami_4.html#E12"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E1:12_1_1_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="nat_d.html#T32" title="NAT_D:th.32">NAT_D:32</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_3"><span class="kw">case </span></a><a NAME="E1:12_1_1_3"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x9</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y9</font> = <font color="Maroon" title="c16">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">s9</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,4); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font></span><br/><a NAME="E3:12_1_1_3"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c19">s9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<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>)
 ;<br/><a NAME="E4:12_1_1_3"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T2" title="EXTPRO_1:th.2">EXTPRO_1:2</a></span>;<br/><a NAME="E5:12_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c19">s9</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="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:12_1_1_3"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="ami_4.html#E12"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><font color="Maroon" title="c15">x</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c17">x9</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c18">y9</font>

<br/>.= 
<font color="Maroon" title="c17">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/><a NAME="E7:12_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c19">s9</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">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:12_1_1_3"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E3:12_1_1_3"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="ami_4.html#E12"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E8:12_1_1_3"/><span class="kw">then </span>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c19">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:12_1_1_3"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="ami_4.html#E11"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E9:12_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c19">s9</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#D9" title="MEMSTR_0:def.9">MEMSTR_0:def 9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">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> <span class="kw"> such that </span><br/><a NAME="E11:12_1_1_3"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c14">P</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c20">k0</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:12_1_1_3"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E7:12_1_1_3"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="ami_4.html#E13"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>;<br/><a NAME="E12:12_1_1_3"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c14">P</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">k0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_1_1_3"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="extpro_1.html#T4" title="EXTPRO_1:th.4">EXTPRO_1:4</a></span>;<br/><a NAME="E13:12_1_1_3"/>
<span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c19">s9</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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:12_1_1_3"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E7:12_1_1_3"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E9:12_1_1_3"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="ami_4.html#E13"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:12_1_1_3"/>
<span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_1_1_3"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="extpro_1.html#T21" title="EXTPRO_1:th.21">EXTPRO_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:12"/>
<span class="p1">(<span class="default"><a href="extpro_1.html#K6" title="EXTPRO_1:func.6">Result</a> (<font color="Maroon" title="c14">P</font>,<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="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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