 Solution

The Jordan–Hölder theorem applies to a module of finite length: ifis a composition series of a module, its successive quotients are simple modules. Any two such composition series of a module have the same length $r$, and their composition factors are pairwise isomorphic after a permutation, with multiplicities preserved. Every finite-dimensional module over $A$ has finite length, since strict inclusions increase vector-space dimension. A finite-dimensional algebra over a field does not by itself make every possible module finite-dimensional, so the finite-length hypothesis should be retained in the statement.