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).