proposition: Lemma tanBank_def : forall x : R, (- MaxBank <= x)%R -> (x <= MaxBank)%R -> cos x <> 0%R.
proof: Proof. intros; cut (MaxBank < PI / 2)%R. intro H1; generalize (Ropp_lt_gt_contravar MaxBank (PI / 2) H1); intro H2; generalize (Rle_lt_trans x MaxBank (PI / 2) H0 H1); intro H3; generalize (Rlt_le_trans (- (PI / 2)) (- MaxBank) x H2 H); intro H4; generalize (cos_gt_0 x H4 H3); intro H5; red in |- *; intro H6; rewrite H6 in H5; elim (Rlt_irrefl 0 H5). apply Rlt_trans with (PI_lb / 2)%R. apply Rlt_trans with 1%R. unfold MaxBank in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 100%R. prove_sup. rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. prove_sup. discrR. unfold PI_lb in |- *; apply Rmult_lt_reg_l with 2%R. prove_sup. rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. prove_sup. discrR. unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2%R. prove_sup. do 2 rewrite (Rmult_comm 2); do 2 rewrite Rmult_assoc; rewrite <- Rinv_l_sym. do 2 rewrite Rmult_1_r; elim PI_approx; intros; assumption. discrR. Qed.
imports: Reals pi_ineq