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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_P1 : forall (f g F G:R -> R) (l a b:R), antiderivative f F a b -> antiderivative g G a b -> antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b.