unfold MaxBank in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ prove_sup | apply Rinv_0_lt_compat; prove_sup ].
apply Rlt_trans with (PI_lb / 4)%R.
unfold MaxBank in |- *; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 100%R.
prove_sup0.
rewrite <- Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; apply Rmult_lt_reg_l with 4%R.
prove_sup.
repeat rewrite (Rmult_comm 4); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym.
unfold PI_lb in |- *; prove_sup.
discrR.
discrR.
unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 4));
apply Rmult_lt_compat_l.
apply Rinv_0_lt_compat; prove_sup.
elim PI_approx; trivial.
Qed.

(**********)
Lemma dtheta_rho :
forall (tr : Trajectory) (t : R),
(- rho (h tr) <= derive_pt (theta tr) t (cond_diff (theta tr) t) <=
rho (h tr))%R.