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(*            Newton's Integral            *)
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Definition Newton_integrable (f:R -> R) (a b:R) : Type :=
  { g:R -> R | antiderivative f g a b \/ antiderivative f g b a }.Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R :=
  let (g,_) := pr in g b - g a.

Lemma FTC_Newton : forall (f:Differential) (a b:R), NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b (FTCN_step1 f a b) = f b - f a.