Let $A,N$ be $A$-modules. I'm trying to prove $A^{\oplus n} \otimes_A N \cong N^{\oplus n}$. First we define $f: F(A,N) \rightarrow N^{\oplus n}$ where $F(A,N)$ is a free module with a basis $e_{a,\nu}, a \in A, \nu \in N$. We see that $f: e_{(a_1,...,a_n),\nu} \mapsto (a_1\nu,...,a_n\nu), a_i \in A, \nu \in N$ is a correctly defined homomorphism, because $e_{\alpha_1(\alpha_2+\alpha_3),\nu}-e_{\alpha_1\alpha_2,\nu}-e_{\alpha_1\alpha_3,\nu} \mapsto (0,...,0), \\ e_{\alpha,\nu_1(\nu_2+\nu_3)}-e_{\alpha,\nu_1\nu_2}-e_{\alpha,\nu_1\nu_3} \mapsto (0,...,0), \\ e_{a\alpha,\nu}- a e_{\alpha,\nu}\mapsto(0,...,0),\\ e_{\alpha,a\nu} - ae_{\alpha,\nu}\mapsto(0,...,0).$
Next we need to show that $f$ is an isomorphism. But how to do that?