Question about isomorphism of direct sums (Step needed to show uniqueness of Elementary Divisors Theorem)

Question

I am trying to proof the uniqueness of Elementary Divisors Theorem, I am stuck in this step: Let $A$ be a principal domain and $M_1, M_2$ two finitely generated $A-$modules. Suppose that

$$A^n \bigoplus T(M_1) \cong A^n \bigoplus T(M_2)$$

where $T(M_1)$ and $T(M_2)$ are the torsion submodule of $M_1$ and $M_2$, respectively. I need to show, from the above isomorphism (whitout using Elementary Divisors Theorem), that $T(M_1) \cong T(M_2)$

Answer 1

Well $T(M_1) \simeq T(A^n\oplus T(M_1))$ and of course if $M\simeq N$ then $T(M)\simeq T(N)$