We define a notion of cut and a proof reduction process for a class of theories, including all equational theories and a first-order formulation of higher-order logic. Proofs normalize for all equational theories. We show that the proof of the normalization theorem for the usual formulation of higher-order logic can be adapted to prove normalization for its first-order formulation. The «hard part» of the proof, that cannot be carried out in higher-order logic itself (the normalization of the system F-omega) is left unchanged. Thus, from the point of view of proof normalization, defining higher-order logic as a different logic or as a first-order theory does not matter. This result also explains a relation between the normalization of propositions and the normalization of proofs in equational theories and in higher-order logic: normalizing propositions does not eliminate cuts, but it transforms them.