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<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="E1:39"/><i><font color="Green" title="E30">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="E2:39"/>
not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ;<br/>

<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</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#D4" target="_self" title="INT_2:def.4">Def4</a></i>;<br/>
<a NAME="E4:39"/>
( <font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">q</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="E5:39"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</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="int_2.html#K3" title="INT_2:func.3">gcd</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#E1:39"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="int_2.html#E3:39"><i><font color="Green" title="E31">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="E6:39"/>
<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/>


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