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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<a NAME="E1:43"/><i><font color="Green" title="E30">A1</font></i>: 
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T1" title="ABSVALUE:th.1">ABSVALUE:1</a></i>;<br/>
<a NAME="E2:43"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<a NAME="E3:43"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="int_2.html#E1:43"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green" title="E32">A3</font></i>: 
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T1" title="ABSVALUE:th.1">ABSVALUE:1</a></i>;<br/>
<a NAME="E5:43"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="int_2.html#E4:43"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E1:43_1"/>
( <font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><a NAME="E2:43_1"/><b>then </b>
( <font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:43"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="int_2.html#E4:43"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:43_1"/>
<font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_2.html#E3:43"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="int_2.html#E6:43"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>


</div>
