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

<span class="kw">set </span><font color="Maroon" title="c1">q</font> =  <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="card_1.html#NK1" title="CARD_1:NK.1">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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">d</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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/>
<span class="kw">take </span><font color="Maroon" title="c3">p</font> = <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="card_1.html#NK1" title="CARD_1:NK.1">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="card_1.html#NK1" title="CARD_1:NK.1">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_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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> <a href="ami_3.html#K13" title="AMI_3:func.13">--&gt;</a> 1,1</span>)</span>; <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="c3">p</font> is  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>  &amp; not <font color="Maroon" title="c3">p</font> is  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  )</span><br/>

<a NAME="E1:82_1_1"/><span class="lab"><font color="Green" title="E18">X</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"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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="ami_1.html#R2" title="AMI_1:pred.2">computes</a>  <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_4.html#T13" title="AMI_4:th.13">AMI_4:13</a></span>;<br/>
<a NAME="E2:82_1_1"/>
<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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> 1,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_4.html#T11" title="AMI_4:th.11">AMI_4:11</a></span>;<br/>
<a NAME="E3:82_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being  <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>  st <br/>( <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</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="card_1.html#NK1" title="CARD_1:NK.1">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="card_1.html#NK1" title="CARD_1:NK.1">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" title="b1">s</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>  &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" title="b1">s</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="card_1.html#NK1" title="CARD_1:NK.1">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="card_1.html#NK1" title="CARD_1:NK.1">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" title="b1">s</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E18">X</font></span></a>, <a class="ref" href="ami_1.html#D30" title="AMI_1:def.30">AMI_1:def 30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:82_1_1"/>
<font color="Maroon" title="c3">p</font> is  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</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="c3">p</font> is  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a> </span><br/>

<span class="kw">take </span>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">p</font> &amp; not  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  )</span><br/>

<a NAME="E5:82_1_1"/>
 <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="card_1.html#NK1" title="CARD_1:NK.1">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> <a href="ami_3.html#K13" title="AMI_3:func.13">--&gt;</a> 1,1</span>)</span> <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="card_1.html#NK1" title="CARD_1:NK.1">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="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E6:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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"><span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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> <a href="ami_3.html#K13" title="AMI_3:func.13">--&gt;</a> 1,1</span>)</span>
 
<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="E7:82_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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="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="card_1.html#NK1" title="CARD_1:NK.1">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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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> <a href="ami_3.html#K13" title="AMI_3:func.13">--&gt;</a> 1,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:82_1_1"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">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="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82_1_1"><span class="lab"><font color="Green" title="E19">A</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span><br/>

<span class="kw">assume </span><a NAME="E9:82_1_1"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2: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">hence </span><a NAME="E10:82_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><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>
