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.

Lemma tan_MaxBank_ub : (tan MaxBank < tan_ub_MaxBank)%R.
generalize tan_MaxBank_approx; intro H; elim H; intros H0 H1; assumption.