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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:2"/><b>then </b><i><font color="Green">E4</font></i>: 
 <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmp_gcd.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="scmp_gcd.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmp_gcd.html#E3:2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmp_gcd.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="ami_4.html#T2">AMI_4:2</a></i>
;<br/>


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