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	ITERATED SUMS
*)

Fixpoint sum_nm (n : nat) : nat -> (nat -> nat) -> nat :=
  fun (m : nat) (f : nat -> nat) =>
  match n with
  | O => f m
  | S n' => f m + sum_nm n' (S m) f
  end.

Lemma sum_nm_times : forall (m n x : nat) (f : nat -> nat), x * sum_nm n m f = sum_nm n m (fun i : nat => x * f i).