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

Lemma is_derive_RInt_param_bound_comp_aux3 :
forall (f : R -> R -> R) a (b : R -> R) (x db : R),
(locally x (fun y : R => ex_RInt (fun t => f y t) a (b x))) ->
(exists eps:posreal, locally x (fun y : R => ex_RInt (fun t => f y t) (b x - eps) (b x + eps))) ->
is_derive b x db ->
(exists eps:posreal, locally x
(fun x0 : R =>
forall t : R,
Rmin a (b x-eps) <= t <= Rmax a (b x+eps) ->
ex_derive (fun u : R => f u t) x0)) ->
(forall t : R,
Rmin a (b x) <= t <= Rmax a (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 (b x)) ->
continuity_pt (fun t => f x t) (b x) ->
is_derive (fun x : R => RInt (fun t => f x t) a (b x)) x
(RInt (fun t : R => Derive (fun u => f u t) x) a (b x) +f x (b x)*db).