Definition primitive (f:R -> R) (a b:R) (h:a <= b)
(pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x)
(x:R) : R :=
match Rle_dec a x with
| left r =>
match Rle_dec x b with
| left r0 => RiemannInt (pr x r r0)
| right _ => f b * (x - b) + RiemannInt (pr b h (Rle_refl b))
end
| right _ => f a * (x - a)
end.

Lemma RiemannInt_P20 :
forall (f:R -> R) (a b:R) (h:a <= b)
(pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x)
(pr0:Riemann_integrable f a b),
RiemannInt pr0 = primitive h pr b - primitive h pr a.