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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:35"/>
not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" target="_self" title="INT_2:def.6">Def6</a></i>;<br/>
<a NAME="E4:35"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">q</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E5:35"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> )
 
<b>by </b><i><a class="txt" href="int_2.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="int_2.html#E3:35"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="int_2.html#D5" target="_self" title="INT_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 ;<br/>


</div>
