proposition: Lemma null_derivative_0 : forall (f:R -> R) (pr:derivable f), constant f -> forall x:R, derive_pt f x (pr x) = 0.
proof: Proof. intros. unfold constant in H. apply derive_pt_eq_0. intros; exists (mkposreal 1 Rlt_0_1); simpl; intros. rewrite (H x (x + h)); unfold Rminus; unfold Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. Qed.
imports: Rbase Rfunctions Ranalysis1 Rtopology Lra