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<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:17"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="ami_4.html#K2" target="_self">Euclide-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E2:17"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="ami_4.html#K2" target="_self">Euclide-Function</a> 
 ;<br/><a NAME="E2:17_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a>  )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E3:17_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> is    <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a>  &amp; <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is    <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a>  )
 <b>by </b><i><a class="ref" href="ami_3.html#T32">AMI_3:32</a></i>;<br/><a NAME="E4:17_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <a href="ami_4.html#K2" target="_self">Euclide-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E1:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ami_4.html#E1:17_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ami_4.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:17_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E4:17"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<a NAME="E5:17"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="ami_4.html#K2" target="_self">Euclide-Function</a> 
 
<b>by </b><i><a class="txt" href="ami_4.html#E4:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ami_4.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:17"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="ami_4.html#K2" target="_self">Euclide-Function</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


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