proposition: Lemma sum_r_derive_sums : forall (An : nat -> C) (r : R) (Pr : finite_cv_radius An r) (z : C), Cnorm z < r -> Cpser (An_deriv An) z (sum_r_derive An r Pr z).
proof: Proof. intros An r Pr z z_bd ; unfold sum_r_derive ; destruct (Rlt_le_dec (Cnorm z) r) as [z_bd2 | Hf]. apply weaksum_r_derive_sums ; apply (middle_is_in_the_middle _ _ z_bd). assert (F : r < r) by (apply Rle_lt_trans with (Cnorm z) ; assumption) ; destruct (Rlt_irrefl _ F). Qed.
imports: Rinterval Ranalysis_def Rpser_def Rpser_base_facts Rpser_cv_facts Rpser_sums Rpow_facts Rsequence_def Rsequence_facts Rseries_def Rseries_facts RFsequence_facts Cpow Cpser_def Cpser_base_facts CFsequence CFsequence_facts Csequence Csequence_def Csequence_facts Cmet Cnorm Ctacfield Cprop_base Canalysis_def Canalysis_cont Canalysis_deriv Canalysis_basic_facts