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<b>let </b><font color="Maroon" title="c13">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c13">p</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>  iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">p</font>,<span class="p2">(<span class="default"><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="c13">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E2:15"/><i><font color="Green" title="E16">A2</font></i>: 
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c13">p</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> 
 ;<br/><a NAME="E2:15_1"/><b>then </b>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <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="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  )
 <b>by </b><i><a class="txt" href="ami_4.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/><a NAME="E3:15_1"/><b>then </b>
( <font color="Maroon" title="c13">p</font> is   <a href="ami_1.html#NM7" title="AMI_1:NM.7">FinPartState</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="Maroon" title="c13">p</font> is   <a href="ami_1.html#NM7" title="AMI_1:NM.7">FinPartState</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  )
 <b>by </b><i><a class="ref" href="ami_1.html#T125" title="AMI_1:th.125">AMI_1:125</a></i>;<br/><a NAME="E4:15_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">p</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#K14" title="AMI_3:func.14">--&gt;</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> &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="c13">p</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#K13" title="AMI_3:func.13">.--&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ami_4.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="ami_4.html#E1:15_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="ami_4.html#D2" target="_self" title="AMI_4:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E5:15_1"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">p</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#K14" title="AMI_3:func.14">--&gt;</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><a NAME="E5:15"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">p</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#K14" title="AMI_3:func.14">--&gt;</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font> )
 ;<br/>

<a NAME="E6:15"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">p</font>,<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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_3.html#K13" title="AMI_3:func.13">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c15">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_4.html#K3" title="AMI_4:func.3">Euclide-Function</a> 
 
<b>by </b><i><a class="txt" href="ami_4.html#E5:15"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="ami_4.html#D2" target="_self" title="AMI_4:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c13">p</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> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


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