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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 antiderivative_P2 : forall (f F0 F1:R -> R) (a b c:R), antiderivative f F0 a b -> antiderivative f F1 b c -> antiderivative f (fun x:R => match Rle_dec x b with | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) a c.