intros; apply Rplus_le_compat_r; apply Rmult_le_compat_l;
[ assumption
| apply sin_incr_1;
[ left; apply Rlt_trans with 0%R;
[ apply _PI2_RLT_0 | apply MinBeta_pos ]
| apply Rle_trans with a; assumption
| left; apply Rlt_trans with 0%R;
[ apply _PI2_RLT_0
| apply Rlt_le_trans with MinBeta; [ apply MinBeta_pos | assumption ] ]
| assumption
| assumption ] ].
Qed.

(**********)
Lemma Math_prop_approx_y_a_PI2 :
forall (a l1 l2 : R) (T : TimeT),
(0 <= a)%R ->
(a <= PI / 2)%R ->
(0 <= l1)%R ->
(l1 <= l2)%R ->
(l1 * sin_lb a + r_ub V * (cos_lb (rho_ub V * MaxT) - 1) <=
l2 * sin a + r_V * (cos (rho_V * T) - 1))%R.