To prove that $a^m.a^n=a^{m+n}$ 
In a group $G$, I want to prove this theorem:
$$\forall a\in G,\; a^m.a^n=a^{m+n},\;m,n\in \Bbb Z.$$

I am thinking that only associative law is sufficient to prove this.please give some suggestions or hint to prove this.thanks in advance

It is clear that $a^{m+1}=a^m.a$. On the other hand, if $a^{m+n}=a^m.a^n$, then$$a^{m+n+1}=a^{m+n}.a=a^m.a^n.a=a^m.a^{n+1}.$$
