Record C1_fun : Type := mkC1
{c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.

(* Specifications: *)
Lemma RiemannInt_P31 :
forall (f:C1_fun) (a b:R),
a <= b -> antiderivative (derive f (diff0 f)) f a b.
Lemma RiemannInt_P32 :
forall (f:C1_fun) (a b:R), Riemann_integrable (derive f (diff0 f)) a b.
Lemma RiemannInt_P33 :
forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b),
a <= b -> RiemannInt pr = f b - f a.
Lemma FTC_Riemann :
forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b),
RiemannInt pr = f b - f a.
(* RiemannInt *)
Lemma RiemannInt_const_bound :
forall f a b l u (h : Riemann_integrable f a b), a <= b ->
(forall x, a < x < b -> l <= f x <= u) ->
l * (b - a) <= RiemannInt h <= u * (b - a).