We prove the following statement: If two
Lagrangians L_1 and L_2 are connected by the
general (invertible) substitution
L_2[\phi]=L_1[f(\phi)]+(something), they result
in the same S-matrix.
This is not a new statement but we could
not find a correct proof (or, at least,
formulation) in the literature. Meanwhile, we
need this theorem to give a functional
formulation of the contraction procedure in
effective theories. Our proof contains two
parts: perturbative and functional. The former
is necessary to provide a foundation for
manipulations with continual integrals made in
the course of the functional proof.