proposition: Lemma derivable_pt_continuity_Riemann_implies_C1 : forall f t0 (pr:forall t t0, Riemann_integrable f t t0), continuity f -> C 1 (fun t => RiemannInt (pr t0 t)).
proof: Proof. intros. assert (derivable (fun t => RiemannInt (pr t0 t))). intro. apply RiemannInt_derivable_pt. apply H. apply (C_S _ _ H0). apply C_0. apply continuity_ext with f. intro. unfold derive. rewrite derive_pt_RiemannInt_1. reflexivity. apply H. apply H. Qed.
imports: PeanoNat Reals Lra RiemannInt Rfunction_classes_def Rextensionality