Divergent Integrals in Renormalizable Field Theories

The proofs of the renormalization of the theories of scalar and pseudoscalar mesons in scalar interaction with nucleons and with the electromagnetic field, given previously by the author, depended on a certain subtraction procedure. A general proof is here given that this subtraction procedure, when applied to a divergent integral in the $S$ matrix of a renormalizable field theory, does, in fact, lead to an absolutely convergent, covariant, and unique remainder.