Section Alpha.
Definition alpha x t := 1/2 * (u0 (x + c * t) + u0 (x - c * t)).
Definition alpha20 x t := 1/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)).
Definition alpha02 x t := c^2/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)).
End Alpha.
Parameter u1 : R -> R.
Hypothesis Du1 : forall x, ex_derive (fun u => u1 u) x.
intros x.
destruct (Du1 x) as (l,Hl).
apply derivable_continuous_pt.
unfold derivable_pt, derivable_pt_abs.
exists l.
now apply is_derive_Reals.
Qed.
intros f a b H.
case (Rle_or_lt a b); intros H1.
apply ex_RInt_Reals_1.
apply continuity_implies_RiemannInt.
exact H1.
intros x _; apply H.
apply ex_RInt_swap.
apply ex_RInt_Reals_1.
apply continuity_implies_RiemannInt.
left; exact H1.
intros x _; apply H.
Qed.

Lemma Iu1: forall a b, ex_RInt (fun u => u1 u) a b.