(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.
(* begin hide *)
Notation Rmin_pos := Rmin_pos (only parsing). (* compat *)
(* end hide *)

Lemma maj_term4 :
forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal)
(f1 f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall a:R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2c ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
l2 <> 0 ->
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
eps / 4.