Let A be a finite dimensional semisimple $k$-algebra, $k$ algebraically closed, and let $S_i$ be a simple module of A with dimension $n_i$. The multiplicity of $S_i$, that is the number of simple submodules of A isomorphic to $S_i$, is equal to $n_i$. Artin-Wedderburn grants us this fact.
Is there a more elementary proof of this statement? I have had little progress trying to find one.
Calculate $\text{Hom}_A(A,S_i)$ in two different ways.
On the one hand, it is isomorphic to $S_i$ as a vector space, so its dimension is $\text{dim }S_i$.
On the other hand, if $A\cong S_1^{n_1}\oplus\dots\oplus S_r^{n_r}$ then by Schur’s lemma its dimension is $n_i$.