How can you prove the fundamental theorem of finitely generated abelian groups using the first isomorphism theorem?

Question

I was able to prove the lemma that lets $G$ be a finitely generated abelian group, generated by $n$ elements $\{g_1,g_2,\dotsc,g_n\}$. Then the homomorphism $: \mathbb Z^n \to G$ defined by $(a_1,a_2,\dotsc,a_n) \mapsto g_1^{a_n} g_2^{a_n}$ is an epimorphism. And I understand the kernel of the epimorphism however I am not sure how to use this lemma and the first isomorphism theorem to prove the fundamental theorem of finitely generated abelian groups. Any suggestions as to the right direction would be helpful.