intro; generalize (Omega_defeq (beta + thetat intr 0) H); intro; rewrite xp0;
replace (- l * cos (thetat intr 0 + beta))%R with
(l * - cos (thetat intr 0 + beta))%R...
apply Rmult_le_pos...
apply l_is_pos...
rewrite <- neg_cos_var; unfold Rminus in |- *; elim H0; intros;
apply cos_ge_0...
apply Rplus_le_reg_l with PI; replace (PI + - (PI / 2))%R with (PI / 2)%R...
rewrite (Rplus_comm PI); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; rewrite <- (Rplus_comm beta)...
pattern PI at 2 in |- *; rewrite double_var; ring...
apply Rplus_le_reg_l with PI; replace (PI + PI / 2)%R with (3 * (PI / 2))%R...
rewrite (Rplus_comm PI); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
rewrite Rplus_0_r; rewrite <- (Rplus_comm beta)...
pattern PI at 2 in |- *; rewrite double_var; ring...
ring...
Qed.

Lemma xpt_Omega :
forall t : R,
(0 <= t)%R ->
(rho_vi * t <= PI)%R ->
Omega (beta + thetat intr 0) = true -> (r_vi * sin (rho_vi * t) <= xp t)%R.