intro t; apply Rlt_trans with (PI / 2)%R...
apply rho_t_PI2...
apply Rlt_le_trans with (PI_ub / 2)%R...
unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2))...
apply Rmult_lt_compat_l...
apply Rinv_0_lt_compat; prove_sup...
assert (H := PI_approx); elim H...
right; unfold PI_ub in |- *; unfold Rdiv in |- *.
change 4%R with (IZR (2 * 2)).
rewrite (mult_IZR 2 2).
rewrite Rmult_assoc;
rewrite <- Rinv_r_sym; [ apply Rmult_1_r | discrR ]...
Qed.

(* Specifications: *)
(**********)
Lemma r_V_is_pos : (0 < r_V)%R.
unfold r_V in |- *; unfold r in |- *; unfold Rdiv in |- *;
apply Rmult_lt_0_compat;
[ apply (TypeSpeed_pos V) | apply Rinv_0_lt_compat; apply rho_pos ].