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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 NewtonInt_P9 : forall (f:R -> R) (a b c:R) (pr1:Newton_integrable f a b) (pr2:Newton_integrable f b c), NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) = NewtonInt f a b pr1 + NewtonInt f b c pr2.