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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="ami_1.html#V12" title="AMI_1:attr.12">-started</a>  <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <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="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="ami_1.html#K11" title="AMI_1:func.11">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  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="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="ami_1.html#K11" title="AMI_1:func.11">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="E1:16"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <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="c14">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="ami_1.html#K11" title="AMI_1:func.11">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="Red">::  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="ami_1.html#K11" title="AMI_1:func.11">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="E2:16"/><span class="lab"><font color="Green" title="E15">A3</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="E3:16"/><span class="lab"><font color="Green" title="E16">A4</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="E4:16"/><span class="lab"><font color="Green" title="E17">A5</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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="E5:16"/><span class="lab"><font color="Green" title="E18">A6</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="#E4:16"><span class="lab"><font color="Green" title="E17">A5</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="16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_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="16_1_1_1"><span class="kw">case </span></a><a NAME="E1:16_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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="E2:16_1_1_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="ami_4.html#E16"><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="Red">::  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="16_1_1_2"><span class="kw">case </span></a><a NAME="E1:16_1_1_2"/><span class="lab"><font color="Green" title="E19">A7</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="Red">::  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="ami_1.html#K11" title="AMI_1:func.11">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="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">s9</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="E3:16_1_1_2"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<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">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="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></span>;<br/><a NAME="E4:16_1_1_2"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c19">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">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="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>
;<br/><a NAME="E6:16_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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:16_1_1_2"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E4:16_1_1_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="ami_4.html#E15"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E7:16_1_1_2"/><span class="kw">then </span>
 <a href="ami_1.html#K6" title="AMI_1:func.6">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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E4:16_1_1_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E8:16_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">P</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> <font color="Maroon" title="c19">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="ami_4.html#E8"><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="ami_1.html#K11" title="AMI_1:func.11">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="ami_1.html#T87" title="AMI_1:th.87">AMI_1:87</a></span>
<br/>.= 
<font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:16_1_1_2"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E4:16_1_1_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="ami_4.html#E15"><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="#E5:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E19">A7</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="Red">::  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="16_1_1_3"><span class="kw">case </span></a><a NAME="E1:16_1_1_3"/><span class="lab"><font color="Green" title="E19">A10</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="Red">::  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="ami_1.html#K11" title="AMI_1:func.11">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="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">s9</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="E3:16_1_1_3"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c19">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">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="E4:16_1_1_3"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<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">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="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></span>;<br/><a NAME="E5:16_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">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> <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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:16_1_1_3"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="ami_4.html#E15"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E6:16_1_1_3"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
 <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="c14">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A2</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="#E1:16_1_1_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/><a NAME="E8:16_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A15</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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E4:16_1_1_3"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E3:16_1_1_3"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="ami_4.html#E15"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a></span>;<br/><a NAME="E9:16_1_1_3"/><span class="kw">then </span>
 <a href="ami_1.html#K6" title="AMI_1:func.6">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="#E1:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:16_1_1_3"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E10:16_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c19">s9</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="ami_1.html#V12" title="AMI_1:attr.12">-started</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D44" title="AMI_1:def.44">AMI_1:def 44</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="E12:16_1_1_3"/><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c14">P</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"><a href="ami_1.html#K10" title="AMI_1:func.10">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="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E1:16_1_1_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E5:16_1_1_3"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E8:16_1_1_3"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E6:16_1_1_3"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="ami_4.html#E16"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>;<br/><a NAME="E13:16_1_1_3"/>
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c20">k0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>;<br/><a NAME="E14:16_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c14">P</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"><a href="ami_1.html#K10" title="AMI_1:func.10">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="#E12:16_1_1_3"><span class="lab"><font color="Green" title="E26">A17</font></span></a></span>;<br/><a NAME="E15:16_1_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="#E3:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E1:16_1_1_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E5:16_1_1_3"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E8:16_1_1_3"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E10:16_1_1_3"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="txt" href="#E6:16_1_1_3"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="ami_4.html#E16"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:16_1_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="#E14:16_1_1_3"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="ami_1.html#T90" title="AMI_1:th.90">AMI_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E7:16"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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