cut (g < 33)%R.
cut (tan MaxBank < 1)%R.
cut (/ h < / 200)%R.
intros; generalize g_pos; intro H2; generalize tan_MaxBank_pos; intro H3;
generalize (Rinv_0_lt_compat h (TypeSpeed_pos h));
intro H4.
generalize
(Rmult_le_0_lt_compat g 33 (tan MaxBank) 1 (Rlt_le 0 g H2)
(Rlt_le 0 (tan MaxBank) H3) H1 H0); intro H5;
generalize (Rmult_lt_0_compat g (tan MaxBank) H2 H3);
intro H6.
rewrite Rmult_1_r in H5.
generalize
(Rmult_le_0_lt_compat (g * tan MaxBank) 33 (/ h) (/ 200)
(Rlt_le 0 (g * tan MaxBank) H6) (Rlt_le 0 (/ h) H4) H5 H).
intro H7; left.
apply Rlt_trans with (33 / 200)%R.
unfold Rdiv in |- *; rewrite <- Rmult_assoc; assumption.
apply Rlt_le_trans with (PI_lb / 2)%R.
apply Rlt_trans with 1%R.
unfold Rdiv in |- *; apply Rmult_lt_reg_l with 200%R.
prove_sup.
rewrite Rmult_1_r; rewrite <- Rmult_comm; rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ prove_sup | discrR ].
apply Rmult_lt_reg_l with 2%R.
prove_sup.
unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; rewrite Rmult_1_l; unfold PI_lb in |- *.
prove_sup.
discrR.
unfold Rdiv in |- *; left; repeat rewrite <- (Rmult_comm (/ 2));
apply Rmult_lt_compat_l.
apply Rinv_0_lt_compat; prove_sup.
elim PI_approx; intros; assumption.
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
prove_sup.
apply TypeSpeed_pos.
apply Rlt_le_trans with MinSpeed.
unfold MinSpeed in |- *; prove_sup.
apply (v_cond1 h).
apply Rlt_trans with tan_ub_MaxBank.
apply tan_MaxBank_ub.
unfold tan_ub_MaxBank in |- *; unfold Rdiv in |- *;
apply Rmult_lt_reg_l with 10%R.
prove_sup.
rewrite <- Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ prove_sup | discrR ].
unfold g in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 10%R.
prove_sup.
rewrite <- Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ prove_sup | discrR ].
Qed.

(**********)
Lemma rho_strict_decreasing :
forall h1 h2 : TypeSpeed, (h1 < h2)%R -> (rho h2 < rho h1)%R.