proposition: Lemma uniqueness_step1 : forall f x l1 l2, limit1_in (fun h => (f (x + h) - f x) / h) (fun h => h <> 0) l1 0 -> limit1_in (fun h => (f (x + h) - f x) / h) (fun h => h <> 0) l2 0 -> l1 = l2.
proof: Proof. intros f x l1 l2 Hl1 Hl2 ; apply (single_limit (fun h => (f (x + h) - f x) / h) (fun h => h <> 0) l1 l2 0); try assumption. intros alpha alpha_pos ; exists (IRC (alpha / 2)) ; split. apply (proj2 (C0_neq_R0_neq _)) ; left ; simpl ; apply Rgt_not_eq ; lra. unfold C_dist ; rewrite Cminus_0_r ; rewrite Cnorm_IRC_Rabs ; rewrite Rabs_right ; lra. 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