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<span class="kw">set </span><font color="Maroon" title="c13">q</font> =  <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">p</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> ;<br/>
<a NAME="E1:20"/><span class="lab"><font color="Green" title="E17">F1</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T119" title="FUNCT_4:th.119">FUNCT_4:119</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ami_1.html#D30" title="AMI_1:def.30">AMI_1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )</span><br/>

<a NAME="E2:20"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
(  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp;  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T57" title="AMI_3:th.57">AMI_3:57</a></span>;<br/>
<a NAME="E3:20"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T57" title="AMI_3:th.57">AMI_3:57</a></span>;<br/>
<a NAME="E4:20"/>
(  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T57" title="AMI_3:th.57">AMI_3:57</a></span>;<br/>
<a NAME="E5:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="ami_4.html#T4" target="_self" title="AMI_4:th.4">Th4</a>, <a class="ref" href="card_1.html#T91" title="CARD_1:th.91">CARD_1:91</a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:20"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a> 
 ; <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">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c17">i2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E8:20"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c16">i1</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="E9:20"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c17">i2</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="E10:20"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K13" title="AMI_3:func.13">--&gt;</a> <font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_4.html#T11" target="_self" title="AMI_4:th.11">Th11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">d</font> = <font color="Maroon" title="c15">x</font> as   <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">t</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E13:20"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T156" title="PBOOLE:th.156">PBOOLE:156</a></span>;<br/>
<a NAME="E14:20"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E15:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E16:20"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E17:20"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
for <font color="Olive" title="b1">t</font> being  <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>   st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">t</font> holds <br/>( <font color="Olive" title="b1">t</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">i1</font> &amp; <font color="Olive" title="b1">t</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="c17">i2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c20">t</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">t</font> implies ( <font color="Maroon" title="c20">t</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">i1</font> &amp; <font color="Maroon" title="c20">t</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="c17">i2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">t</font>
 ; <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="c20">t</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">i1</font> &amp; <font color="Maroon" title="c20">t</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="c17">i2</font> )</span><br/>

<a NAME="E3:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c20">t</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> = 
<font color="Maroon" title="c18">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="txt" href="#E16:20"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
<br/>.= 
<font color="Maroon" title="c16">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/> <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="c20">t</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="c17">i2</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c20">t</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> = 
<font color="Maroon" title="c18">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
<br/>.= 
<font color="Maroon" title="c17">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</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>
<a NAME="E18:20"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E19:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_4.html#T4" target="_self" title="AMI_4:th.4">Th4</a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:20_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:20_2"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 5 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E6:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A18</font></span>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T91" title="CARD_1:th.91">CARD_1:91</a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:20_2"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E4:20_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:20_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="ami_3.html#T56" title="AMI_3:th.56">AMI_3:56</a>, <a class="ref" href="ami_3.html#T57" title="AMI_3:th.57">AMI_3:57</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>
<a NAME="E21:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T33" title="FUNCT_4:th.33">FUNCT_4:33</a></span>;<br/>
<a NAME="E22:20"/>
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E23:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E24:20"/>
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E25:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A21</font></span>: 
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E26:20"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></span>
;<br/>
<a NAME="E28:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</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="txt" href="#E26:20"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="ami_1.html#D28" title="AMI_1:def.28">AMI_1:def 28</a></span>;<br/>
<a NAME="E29:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c19">t</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="txt" href="#E13:20"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="ami_1.html#T92" title="AMI_1:th.92">AMI_1:92</a></span>;<br/>
<a NAME="E30:20"/>
 <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> <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></span>;<br/>
<a NAME="E31:20"/><span class="kw">then </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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E32:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A25</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T209" title="RELAT_1:th.209">RELAT_1:209</a></span>;<br/>
<a NAME="E33:20"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> is  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c20">A</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>,<span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">C</font> = 5;<br/>
<span class="kw">let </span><font color="Maroon" title="c22">s1</font>, <font color="Maroon" title="c23">s2</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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ami_1.html#D25" title="AMI_1:def.25">AMI_1:def 25</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">s1</font> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">s2</font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_4"/><span class="lab"><font color="Green" title="E39">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">s1</font>
 <span class="kw">and </span><br/><a NAME="E2:20_4"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">s2</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:20_4"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
( <font color="Maroon" title="c23">s2</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">i1</font> &amp; <font color="Maroon" title="c23">s2</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="c17">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:20_4"><span class="lab"><font color="Green" title="E40">A28</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c24">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span></span><br/>

<a NAME="E4:20_4"/><span class="lab"><font color="Green" title="E42">NX1</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></span>;<br/>
<a NAME="E5:20_4"/><span class="lab"><font color="Green" title="E43">NX2</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></span>;<br/>
<a NAME="E6:20_4"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="relat_1.html#T105" title="RELAT_1:th.105">RELAT_1:105</a></span>;<br/>
<a NAME="E7:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A30</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,$1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,$1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,$1</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> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,$1</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> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,$1</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,$1</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> 1</span>)</span> );<br/>
<a NAME="E8:20_4"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
(  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s1</font> &amp;  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s2</font> )
 
<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="E9:20_4"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c22">s1</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="txt" href="#E28:20"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="ami_1.html#T92" title="AMI_1:th.92">AMI_1:92</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A34</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/>
<a NAME="E11:20_4"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="relat_1.html#T105" title="RELAT_1:th.105">RELAT_1:105</a></span>;<br/>
<a NAME="E12:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A35</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A36</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5 = 
 <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_4.html#T4" target="_self" title="AMI_4:th.4">Th4</a>, <a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E42">NX1</font></span></a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E5:20_4"><span class="lab"><font color="Green" title="E43">NX2</font></span></a>, <a class="ref" href="ami_4.html#T4" target="_self" title="AMI_4:th.4">Th4</a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></span>
;<br/>
<a NAME="E14:20_4"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c23">s2</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="txt" href="#E28:20"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E2:20_4"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="ami_1.html#T92" title="AMI_1:th.92">AMI_1:92</a></span>;<br/>
<a NAME="E15:20_4"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">i</font>] &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 holds <br/><font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">i</font>, <font color="Maroon" title="c26">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">S<sub>2</sub></font>[4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font>] &amp; <font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 implies <font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_4_3"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:20_4_3"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:20_4_3"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 or <font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>] )</span><br/>

<a NAME="E4:20_4_3"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:20_4_3"/>
( <font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>]</span><br/>

<a NAME="E6:20_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: 
( <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_3_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="E1:20_4_3_1"/>
(  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E14:20_4"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E4:20_4_3"><span class="lab"><font color="Green" title="E55">A42</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_3_1_1"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>]</span><br/><div class="add"><a NAME="E2:20_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></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="#E1:20_4_3"><span class="lab"><font color="Green" title="E52">A39</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A46</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E2:20_4_3"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E1:20_4_3_1_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
;<br/><span class="lab"><font color="Green" title="E60">A47</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E2:20_4_3_1_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E3:20_4_3"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E1:20_4_3_1_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
;<br/><span class="lab"><font color="Green" title="E61">A48</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E2:20_4_3_1_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E3:20_4_3"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E1:20_4_3_1_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>
;<br/><a NAME="E6:20_4_3_1_1"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:20_4_3_1_1"/><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:20_4_3_1_1"/><span class="lab"><font color="Green" title="E64">A51</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:20_4_3_1_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>;<br/><a NAME="E9:20_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A52</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>;<br/><a NAME="E10:20_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A53</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>;<br/><a NAME="E11:20_4_3_1_1"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E2:20_4_3_1_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="ami_4.html#T5" target="_self" title="AMI_4:th.5">Th5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A55</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E8:20_4_3_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E4:20_4_3_1_1"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
;<br/><span class="lab"><font color="Green" title="E69">A56</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E11:20_4_3_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E8:20_4_3_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E3:20_4_3_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E5:20_4_3_1_1"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
;<br/><a NAME="E14:20_4_3_1_1"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E11:20_4_3_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>;<br/><a NAME="E15:20_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A58</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>;<br/><span class="lab"><font color="Green" title="E72">A59</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E11:20_4_3_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E6:20_4_3_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E8:20_4_3_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E3:20_4_3_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E5:20_4_3_1_1"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="ami_4.html#T6" target="_self" title="AMI_4:th.6">Th6</a></span>
;<br/><a NAME="E17:20_4_3_1_1"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><span class="lab"><font color="Green" title="E74">A61</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E14:20_4_3_1_1"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E9:20_4_3_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E12:20_4_3_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>
;<br/><span class="lab"><font color="Green" title="E75">A62</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E14:20_4_3_1_1"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E7:20_4_3_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E9:20_4_3_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E13:20_4_3_1_1"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="ref" href="ami_4.html#T7" target="_self" title="AMI_4:th.7">Th7</a></span>
;<br/><a NAME="E20:20_4_3_1_1"/>
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E21:20_4_3_1_1"/><span class="kw">then </span>
( (  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp;  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) or (  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></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="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E17:20_4_3_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E15:20_4_3_1_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E10:20_4_3_1_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E19:20_4_3_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="ami_4.html#T8" target="_self" title="AMI_4:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E22:20_4_3_1_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4_3"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E11:20_4_3_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E8:20_4_3_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E14:20_4_3_1_1"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E9:20_4_3_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E15:20_4_3_1_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E10:20_4_3_1_1"><span class="lab"><font color="Green" title="E66">A53</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"> ( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> )</span><br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E17:20_4_3_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E15:20_4_3_1_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="ami_4.html#T8" target="_self" title="AMI_4:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E17:20_4_3_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E10:20_4_3_1_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E18:20_4_3_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="ami_4.html#T8" target="_self" title="AMI_4:th.8">Th8</a></span>
;<br/><span class="kw">hence </span><a NAME="E24:20_4_3_1_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#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="txt" href="#E6:20_4_3"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E3:20_4_3_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E16:20_4_3_1_1"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E18:20_4_3_1_1"><span class="lab"><font color="Green" title="E74">A61</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E17:20_4_3_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E15:20_4_3_1_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="ami_4.html#T8" target="_self" title="AMI_4:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E17:20_4_3_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E10:20_4_3_1_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E19:20_4_3_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="ami_4.html#T8" target="_self" title="AMI_4:th.8">Th8</a></span>
;<br/><span class="kw">hence </span><a NAME="E26:20_4_3_1_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4_3"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E5:20_4_3_1_1"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E13:20_4_3_1_1"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="txt" href="#E19:20_4_3_1_1"><span class="lab"><font color="Green" title="E75">A62</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="20_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_3_1_2"/><span class="lab"><font color="Green" title="E57">A63</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>]</span><br/><div class="add"><a NAME="E2:20_4_3_1_2"/><span class="kw">then </span>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E1:20_4_3"><span class="lab"><font color="Green" title="E52">A39</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/><a NAME="E3:20_4_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A64</font></span>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></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> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T89" title="AMI_1:th.89">AMI_1:89</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:20_4_3_1_2"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:20_4_3_1_2"><span class="lab"><font color="Green" title="E57">A63</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><a NAME="E5:20_4_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_3"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E2:20_4_3"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E3:20_4_3"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E3:20_4_3_1_2"><span class="lab"><font color="Green" title="E58">A64</font></span></a>, <a class="ref" href="ami_1.html#T89" title="AMI_1:th.89">AMI_1:89</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c22">s1</font>
<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 href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="ami_1.html#D28" title="AMI_1:def.28">AMI_1:def 28</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_4"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="ami_1.html#D28" title="AMI_1:def.28">AMI_1:def 28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</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="E17:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A65</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E8:20_4"><span class="lab"><font color="Green" title="E45">A32</font></span></a></span>;<br/>
<a NAME="E18:20_4"/><span class="lab"><font color="Green" title="E53">A66</font></span>: 
4 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E19:20_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c24">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_d.html#S2" title="NAT_D:sch.2">NAT_D:sch 2</a>(<a class="txt" href="#E17:20_4"><span class="lab"><font color="Green" title="E52">A65</font></span></a>, <a class="txt" href="#E18:20_4"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="txt" href="#E15:20_4"><span class="lab"><font color="Green" title="E51">A38</font></span></a>);<br/></span>
<a NAME="E20:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A67</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>,<span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>,<span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_4"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="ref" href="grfunc_1.html#T92" title="GRFUNC_1:th.92">GRFUNC_1:92</a></span>;<br/>
<a NAME="E21:20_4"/>
5 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E22:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">XX</font></span>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> 5 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> 5</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> 5
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>,<span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> 5
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T42" title="ENUMSET1:th.42">ENUMSET1:42</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E24:20_4"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">s1</font></span>)</span>,<font color="Maroon" title="c22">s1</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c23">s2</font></span>)</span>,<font color="Maroon" title="c23">s2</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_4"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E20:20_4"><span class="lab"><font color="Green" title="E54">A67</font></span></a>, <a class="txt" href="#E22:20_4"><span class="lab"><font color="Green" title="E55">XX</font></span></a>, <a class="ref" href="relat_1.html#T185" title="RELAT_1:th.185">RELAT_1:185</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>
<span class="kw">take </span>
<font color="Maroon" title="c18">d</font>
; <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="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E34:20"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</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"><span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> )</span><br/>

<a NAME="E35:20"/><span class="lab"><font color="Green" title="E39">A68</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> is  <a href="ami_1.html#V11" title="AMI_1:attr.11">halting</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c20">i19</font> = <font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c21">i29</font> = <font color="Maroon" title="c17">i2</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="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">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">let </span><font color="Maroon" title="c22">t</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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">t</font> or  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c23">t9</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4;<br/>
<span class="kw">assume </span><a NAME="E2:20_5"/><span class="lab"><font color="Green" title="E40">A69</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font></span><br/>

<a NAME="E3:20_5"/><span class="kw">then </span>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T105" title="RELAT_1:th.105">RELAT_1:105</a></span>;<br/>
<a NAME="E4:20_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A70</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:20_5"/><span class="lab"><font color="Green" title="E42">A71</font></span>: 
<font color="Maroon" title="c22">t</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="c17">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:20_5"><span class="lab"><font color="Green" title="E40">A69</font></span></a></span>;<br/>
<a NAME="E6:20_5"/><span class="lab"><font color="Green" title="E43">A72</font></span>: 
( <font color="Maroon" title="c22">t</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>  &amp; <font color="Maroon" title="c22">t</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">i1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:20_5"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="ref" href="ami_1.html#T92" title="AMI_1:th.92">AMI_1:92</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_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="E1:20_5_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">i2</font> or <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> or <font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i2</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="20_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_1"/>
<font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font></span><br/><div class="add"><a NAME="E2:20_5_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,<font color="Olive" title="b1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</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="E3:20_5_1_1"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T83" title="AMI_1:th.83">AMI_1:83</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="20_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2"/><span class="lab"><font color="Green" title="E44">A73</font></span>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E45">A74</font></span>: <font color="Maroon" title="c16">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c17">i2</font> = 
<font color="Maroon" title="c20">i19</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c21">i29</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="#E1:20_5_1_2"><span class="lab"><font color="Green" title="E44">A73</font></span></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>
;<br/><a NAME="E3:20_5_1_2"/><span class="lab"><font color="Green" title="E46">A75</font></span>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</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="E4:20_5_1_2"/>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</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="E5:20_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<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>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E3:20_5_1_2"><span class="lab"><font color="Green" title="E46">A75</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="E6:20_5_1_2"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E2:20_5_1_2"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="txt" href="#E3:20_5_1_2"><span class="lab"><font color="Green" title="E46">A75</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="E7:20_5_1_2"/><span class="kw">then </span>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</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><a NAME="E8:20_5_1_2"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T83" title="AMI_1:th.83">AMI_1:83</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="20_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_3"/><span class="lab"><font color="Green" title="E44">A76</font></span>: 
<font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font></span><br/><div class="add"><a NAME="E2:20_5_1_3"/><span class="lab"><font color="Green" title="E45">A77</font></span>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</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="E3:20_5_1_3"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></span>;<br/><a NAME="E4:20_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A78</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a></span>;<br/><a NAME="E5:20_5_1_3"/><span class="lab"><font color="Green" title="E47">A79</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</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/><font color="Maroon" title="c16">i1</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c17">i2</font> = 
<font color="Maroon" title="c20">i19</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c21">i29</font>

<br/>.= 
<font color="Maroon" title="c20">i19</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_3"><span class="lab"><font color="Green" title="E44">A76</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/><a NAME="E7:20_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A80</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E5:20_5_1_3"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E2:20_5_1_3"><span class="lab"><font color="Green" title="E45">A77</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="E8:20_5_1_3"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</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="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E2:20_5_1_3"><span class="lab"><font color="Green" title="E45">A77</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="E9:20_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A81</font></span>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4 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/><a NAME="E10:20_5_1_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</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="c17">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="txt" href="#E6:20_5"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E5:20_5"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E5:20_5_1_3"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E2:20_5_1_3"><span class="lab"><font color="Green" title="E45">A77</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">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="E12:20_5_1_3"/><span class="lab"><font color="Green" title="E50">A82</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span> <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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p4">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p3">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span>,<font color="Maroon" title="c24">k0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:20_5_1_3"><span class="lab"><font color="Green" title="E44">A76</font></span></a>, <a class="txt" href="#E7:20_5_1_3"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="txt" href="#E9:20_5_1_3"><span class="lab"><font color="Green" title="E49">A81</font></span></a>, <a class="txt" href="#E4:20_5_1_3"><span class="lab"><font color="Green" title="E46">A78</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="E13:20_5_1_3"/><span class="lab"><font color="Green" title="E51">T</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></span>;<br/><a NAME="E14:20_5_1_3"/>
 <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">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> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,4</span>)</span>,<font color="Maroon" title="c24">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="E15:20_5_1_3"/><span class="kw">then </span>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</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> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font></span>)</span>,<font color="Maroon" title="c22">t</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">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="#E12:20_5_1_3"><span class="lab"><font color="Green" title="E50">A82</font></span></a>, <a class="txt" href="#E13:20_5_1_3"><span class="lab"><font color="Green" title="E51">T</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:20_5_1_3"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c22">t</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T83" title="AMI_1:th.83">AMI_1:83</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="E36:20"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of  <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E17">F1</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"> <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span><br/>

<a NAME="E37:20"/><span class="lab"><font color="Green" title="E40">A83</font></span>: 
 <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">t</font></span>)</span>,<font color="Maroon" title="c19">t</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a></span>;<br/>
<a NAME="E38:20"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 
;<br/>
<a NAME="E39:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A84</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">t</font></span>)</span>,<font color="Maroon" title="c19">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E40:20"/>
 <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> <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></span>;<br/>
<a NAME="E41:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A85</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E42:20"/><span class="lab"><font color="Green" title="E43">A86</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/>
<a NAME="E43:20"/><span class="lab"><font color="Green" title="E44">A87</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c17">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/>
<a NAME="E44:20"/><span class="lab"><font color="Green" title="E45">A88</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></span>;<br/>
<a NAME="E45:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A89</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/>
<a NAME="E46:20"/><span class="lab"><font color="Green" title="E47">A90</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E44:20"><span class="lab"><font color="Green" title="E45">A88</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E47:20"/><span class="lab"><font color="Green" title="E48">A91</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E48:20"/><span class="kw">then </span>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E49:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A92</font></span>: 
<font color="Maroon" title="c19">t</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="c17">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:20"><span class="lab"><font color="Green" title="E47">A90</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E44">A87</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/>
<a NAME="E50:20"/>
 <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="c19">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E41:20"><span class="lab"><font color="Green" title="E42">A85</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E51:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">XX</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>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T209" title="RELAT_1:th.209">RELAT_1:209</a></span>;<br/>
<a NAME="E52:20"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:20"><span class="lab"><font color="Green" title="E48">A91</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E53:20"/><span class="kw">then </span>
<font color="Maroon" title="c19">t</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">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:20"><span class="lab"><font color="Green" title="E47">A90</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E43">A86</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/>
<a NAME="E54:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A93</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">t</font></span>)</span>,<font color="Maroon" title="c19">t</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="c16">i1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c17">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E49:20"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="ami_4.html#T10" target="_self" title="AMI_4:th.10">Th10</a>, <a class="txt" href="#E51:20"><span class="lab"><font color="Green" title="E50">XX</font></span></a></span>;<br/>
<a NAME="E55:20"/>
(  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  &amp;  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T57" title="AMI_3:th.57">AMI_3:57</a></span>;<br/>
<a NAME="E56:20"/><span class="kw">then </span>
( not  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E57:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">K</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:20"><span class="lab"><font color="Green" title="E48">A91</font></span></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_6"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:20_6"/><span class="lab"><font color="Green" title="E53">W1</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 <span class="kw">and </span><br/><a NAME="E4:20_6"/><span class="lab"><font color="Green" title="E54">W2</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:20_6"/><span class="lab"><font color="Green" title="E55">A</font></span>: 
<font color="Maroon" title="c20">x</font> is    <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="#E4:20_6"><span class="lab"><font color="Green" title="E54">W2</font></span></a></span>;<br/><a NAME="E6:20_6"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_6"><span class="lab"><font color="Green" title="E53">W1</font></span></a>, <a class="txt" href="#E47:20"><span class="lab"><font color="Green" title="E48">A91</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:20_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_6"><span class="lab"><font color="Green" title="E55">A</font></span></a>, <a class="ref" href="ami_3.html#T56" title="AMI_3:th.56">AMI_3:56</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>
<a NAME="E59:20"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:20"><span class="lab"><font color="Green" title="E52">K</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/>
<a NAME="E60:20"/><span class="kw">then </span>
<font color="Maroon" title="c18">d</font> is  <a href="ami_1.html#V14" title="AMI_1:attr.14">data-only</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D51" title="AMI_1:def.51">AMI_1:def 51</a></span>;<br/>
<a NAME="E61:20"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T155" title="AMI_1:th.155">AMI_1:155</a></span>;<br/>
<a NAME="E62:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">x2</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T76" title="FUNCT_4:th.76">FUNCT_4:76</a></span>;<br/>
<a NAME="E63:20"/><span class="lab"><font color="Green" title="E54">x3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c17">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<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><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E64:20"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c17">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:20"><span class="lab"><font color="Green" title="E48">A91</font></span></a>, <a class="txt" href="#E63:20"><span class="lab"><font color="Green" title="E54">x3</font></span></a>, <a class="ref" href="zfmisc_1.html#T12" title="ZFMISC_1:th.12">ZFMISC_1:12</a></span>;<br/>
<a NAME="E65:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A94</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c17">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E46">A89</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E66:20"/>
<a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c17">i2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="ami_4.html#T12" target="_self" title="AMI_4:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E67:20"/>
<a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:20"><span class="lab"><font color="Green" title="E41">A84</font></span></a>, <a class="txt" href="#E54:20"><span class="lab"><font color="Green" title="E51">A93</font></span></a>, <a class="txt" href="#E65:20"><span class="lab"><font color="Green" title="E55">A94</font></span></a>, <a class="txt" href="#E37:20"><span class="lab"><font color="Green" title="E40">A83</font></span></a>, <a class="txt" href="#E62:20"><span class="lab"><font color="Green" title="E53">x2</font></span></a>, <a class="ref" href="funct_4.html#T90" title="FUNCT_4:th.90">FUNCT_4:90</a>, <a class="ref" href="relat_1.html#T186" title="RELAT_1:th.186">RELAT_1:186</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>
