intros; unfold Rminus in |- *; apply Rplus_le_compat_l;
apply Ropp_ge_le_contravar; apply Rle_ge; apply Rmult_le_compat_l;
[ assumption
| apply cos_decr_1;
[ left; apply MinBeta_pos
| apply Rle_trans with a;
[ assumption
| left; apply Rle_lt_trans with (PI / 2)%R;
[ assumption | apply PI2_Rlt_PI ] ]
| left; apply Rlt_le_trans with MinBeta;
[ apply MinBeta_pos | assumption ]
| left; apply Rle_lt_trans with (PI / 2)%R;
[ assumption | apply PI2_Rlt_PI ]
| assumption ] ].
Qed.

(**********)
Lemma Math_prop_approx_x_a_PI2 :
forall (a l1 l2 : R) (T : TimeT),
(0 <= a)%R ->
(a <= PI / 2)%R ->
(0 <= l1)%R ->
(l1 <= l2)%R ->
(r_lb V * sin_lb (rho_lb V * MinT) - l2 * cos_ub a <=
r_V * sin (rho_V * T) - l1 * cos a)%R.