proposition: Lemma uniqueness_step3 : forall f x l, limit1_in (fun h => (f (x + h) - f x) / h) (fun h => h <> 0) l 0 -> derivable_pt_lim f x l.
proof: Proof. unfold limit1_in, derivable_pt_lim, limit_in, C_dist ; simpl ; unfold C_dist ; intros f x l Hf_lim eps eps_pos ; destruct (Hf_lim eps eps_pos) as (alpha, [alpha_pos Halpha]) ; clear Hf_lim ; exists (mkposreal alpha alpha_pos) ; intros h h_neq Hyp ; apply Halpha ; rewrite Cminus_0_r ; split ; assumption. Qed.
imports: MyRIneq Rinterval Ranalysis_def Ranalysis_def_simpl Ranalysis_continuity Ranalysis_derivability Ranalysis_usual Ranalysis_facts Ranalysis_MVT Canalysis_diff Cprop_base Cnorm Ctacfield Cmet Canalysis_def Cderiv