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.