Jordan–Hölder theorem for modules

Every two finite composition series of a module have the same multiset of simple composition factors, up to isomorphism and order. One proves this by induction on the length: if the first simple submodules differ, their intersection is zero, and the quotient by their sum identifies the remaining common comparison problem; if they coincide, pass directly to the quotient. The theorem makes the multiplicity of a simple module independent of a chosen composition series. It does not assert that the module is a direct sum of its composition factors.