proposition: 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.
proof: Proof. intros. assert (H7 := H3 h H6). assert (H8 := H2 h H4 H5). apply Rle_lt_trans with (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)). - rewrite Rabs_mult. apply Rmult_le_compat_r. + apply Rabs_pos. + rewrite Rabs_inv; left; exact H7. - apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)). + apply Rmult_lt_compat_l. * unfold Rdiv; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. * exact H8. + right; unfold Rdiv. repeat rewrite Rabs_mult. rewrite Rabs_inv. rewrite (Rabs_pos_eq 8) by now apply IZR_le. rewrite (Rabs_pos_eq eps). * field. now apply Rabs_no_R0. * now apply Rlt_le. Qed.
imports: Rbase Rfunctions Ranalysis1 Lra