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<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E6</font></i>: 
 <a href="ami_4.html#K1" target="_self">Euclide-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E7</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0
 ;<br/>

<i><font color="Green">E8</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="ami_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T55">AMI_1:55</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E1:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="ami_4.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ami_4.html#E6"><i><font color="Green">Lemma4</font></i></a></i>
;<br/>
<b>thus </b> <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span>

<br/>.= 
 <a href="ami_3.html#K11">Next</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E3:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ami_3.html#T8">AMI_3:8</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="ami_3.html#T54">AMI_3:54</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> 1

;<br/>

<b>thus </b><a NAME="E5:7"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E3:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ami_3.html#T8">AMI_3:8</a></i>;<br/>

<b>thus </b><a NAME="E6:7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E3:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ami_3.html#T8">AMI_3:8</a></i>;<br/>


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