(* begin hide *)
Notation Rmin_pos := Rmin_pos (only parsing).

(* Specifications: *)
Lemma maj_term1 : forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal) (f1 f2:R -> R), 0 < eps -> f2 x <> 0 -> f2 (x + h) <> 0 -> (forall h:R, h <> 0 -> Rabs h < alp_f1d -> Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) -> (forall a:R, Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> h <> 0 -> Rabs h < alp_f1d -> Rabs h < Rmin eps_f2 alp_f2 -> Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
Lemma maj_term2 : forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal) (f2:R -> R), 0 < eps -> f2 x <> 0 -> f2 (x + h) <> 0 -> (forall a:R, Rabs a < alp_f2t2 -> Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) -> (forall a:R, Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> h <> 0 -> Rabs h < alp_f2t2 -> Rabs h < Rmin eps_f2 alp_f2 -> l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
Lemma maj_term3 : forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal) (f1 f2:R -> R), 0 < eps -> f2 x <> 0 -> f2 (x + h) <> 0 -> (forall h:R, h <> 0 -> Rabs h < alp_f2d -> Rabs ((f2 (x + h) - f2 x) / h - l2) < Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) -> (forall a:R, Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> h <> 0 -> Rabs h < alp_f2d -> Rabs h < Rmin eps_f2 alp_f2 -> f1 x <> 0 -> Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) < eps / 4.
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.
Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
Lemma Rabs_4 : forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
Lemma Rlt_4 : forall a b c d e f g h:R, a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.