(** * Definitions on [R] *)

Definition Derive (f : R -> R) (x : R) := real (Lim (fun h => (f (x+h) - f x)/h) 0).

Lemma fn_eq_Derive_eq: forall f g a b, continuity_pt f a -> continuity_pt f b -> continuity_pt g a -> continuity_pt g b -> (forall x, a < x < b -> ex_derive f x) -> (forall x, a < x < b -> ex_derive g x) -> (forall x, a < x < b -> Derive f x = Derive g x) -> exists C, forall x, a <= x <= b -> f x = g x + C.