End PROD'.
Fixpoint composeOptN (A: Type) (f: A -> option A)
(n: nat) (w: A) {struct n} : option A :=
match n  with
| S n' => match f w with Some w' => composeOptN A f n' w' | None => None end
| O => Some w
end.
Arguments composeOptN [A] _ _ _.
Definition ageN {A} `{ageable A}: nat -> A -> option A := composeOptN age1.

Lemma ageN_compose {A} `{agA : ageable A}:
forall a b c phi1 phi2 phi3,ageN a phi1 = Some phi2 ->
ageN b phi2 = Some phi3 ->  (a+b=c)%nat ->  ageN c phi1 = Some phi3.