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).