Section RInt_comp.
Context {V : NormedModule R_AbsRing}.
End RInt_comp.
(** * Parametric integrals *)

Lemma is_derive_RInt_param_bound_comp :
forall (f : R -> R -> R) (a b : R -> R) (x da db : R),
(locally x (fun y : R => ex_RInt (fun t => f y t) (a x) (b x))) ->
(exists eps:posreal, locally x (fun y : R => ex_RInt (fun t => f y t) (a x - eps) (a x + eps))) ->
(exists eps:posreal, locally x (fun y : R => ex_RInt (fun t => f y t) (b x - eps) (b x + eps))) ->
is_derive a x da ->
is_derive b x db ->
(exists eps:posreal, locally x
(fun x0 : R =>
forall t : R,
Rmin (a x-eps) (b x -eps) <= t <= Rmax (a x+eps) (b x+eps) ->
ex_derive (fun u : R => f u t) x0)) ->
(forall t : R,
Rmin (a x) (b x) <= t <= Rmax (a x) (b x) ->
continuity_2d_pt (fun u v : R => Derive (fun z : R => f z v) u) x t) ->
(locally_2d (fun x' t =>
continuity_2d_pt (fun u v : R => Derive (fun z : R => f z v) u) x' t) x (a x)) ->
(locally_2d (fun x' t =>
continuity_2d_pt (fun u v : R => Derive (fun z : R => f z v) u) x' t) x (b x)) ->
continuity_pt (fun t => f x t) (a x) ->
continuity_pt (fun t => f x t) (b x) ->
is_derive (fun x : R => RInt (fun t => f x t) (a x) (b x)) x
(RInt (fun t : R => Derive (fun u => f u t) x) (a x) (b x)+(-f x (a x))*da+(f x (b x))*db).