End Beta.
Hypothesis f : R -> R -> R.
Section Gamma.
Definition gamma x t := 1/(2*c) * RInt (fun tau => RInt (fun xi => f xi tau) (x - c * (t - tau)) (x + c * (t - tau))) 0 t.
Definition gamma20 x t := 1/(2*c) * RInt (fun tau => Derive (fun u => f u tau) (x + c * (t - tau)) - Derive (fun u => f u tau) (x - c * (t - tau))) 0 t.
Definition gamma02 x t := (f x t + c/2 * RInt (fun tau => Derive (fun u => f u tau) (x + c * (t - tau)) - Derive (fun u => f u tau) (x - c * (t - tau))) 0 t).

(* Specifications: *)
Lemma gamma20_lim :
forall x t, is_derive_n (fun u => gamma u t) 2 x (gamma20 x t).
Lemma gamma02_lim :
forall x t, is_derive_n (fun u => gamma x u) 2 t (gamma02 x t).