There's a proposition that states that for an $R$-module $E=E_1^{(n_1)}\oplus\cdots\oplus E_r^{(n_r)}$, the $E_i$ being nonisomorphic and each $E_i$ being repeated $n_i$ times, then the $E_i$ are uniquely determined up to isomorphism, and the multiplicities are uniquely determined.
The proof starts by assuming there is an isomorphism between direct sum decompositions into simple modules $ E_1^{(n_1)}\oplus\cdots\oplus E_r^{(n_r)}\to F_1^{(m_1)}\oplus\cdots\oplus F_s^{(m_s)} $ with $E_i$ nonisomorphic, and $F_j$ nonisomorphic. It then states from Schur's lemma, (that every nonzero homomorphism between simple modules is an isomorphism), that we conclude that each $E_i$ is isomorphic to some $F_j$, and conversely. I don't see how it applies here. How is Schur's lemma used to get that conclusion in the proof?