prove_sup.
rewrite Rmult_0_r; rewrite <- Rmult_comm; rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ prove_sup | discrR ].
Qed.

(**********)
Lemma tan_ub_MaxBank_pos : (0 < tan_ub_MaxBank)%R.
apply Rlt_trans with tan_lb_MaxBank;
[ apply tan_lb_MaxBank_pos
| apply Rlt_trans with (tan MaxBank);
[ apply tan_MaxBank_lb | apply tan_MaxBank_ub ] ].