Proof with trivial.
intros...
rewrite Dyp...
generalize (theta_inv t H); intro; elim H1; intros...
cut (thetat intr t - thetat intr 0 <= rho_vi * t)%R...
cut (- (rho_vi * t) <= thetat intr t - thetat intr 0)%R...
intros...
split...
replace (- (vi intr * sin (rho_vi * t)))%R with
(vi intr * - sin (rho_vi * t))%R...
apply Rmult_le_compat_l...
left; apply TypeSpeed_pos...
rewrite <- (sin_neg (rho_vi * t))...
apply sin_incr_1...
apply Ropp_ge_le_contravar; apply Rle_ge...
apply Rle_trans with 0%R...
rewrite <- Ropp_0...
apply Ropp_ge_le_contravar; apply Rle_ge; apply (rho_t_pos t H)...
left; apply PI2_RGT_0...
apply Rle_trans with (- (rho_vi * t))%R...
apply Ropp_ge_le_contravar...
apply Rle_ge...
apply Rle_trans with (rho_vi * t)%R...
ring...
apply Rmult_le_compat_l...
left; apply TypeSpeed_pos...
apply sin_incr_1...
apply Rle_trans with (- (rho_vi * t))%R...
apply Ropp_ge_le_contravar; apply Rle_ge...
apply Rle_trans with (rho_vi * t)%R...
apply Rle_trans with 0%R...
rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; left;
apply PI2_RGT_0...
apply (rho_t_pos t H)...
apply Rplus_le_reg_l with (thetat intr 0)...
unfold Rminus in |- *...
rewrite (Rplus_comm (thetat intr t))...
repeat rewrite <- Rplus_assoc...
rewrite Rplus_opp_r...
rewrite Rplus_0_l...
apply Rplus_le_reg_l with (thetat intr 0)...
unfold Rminus in |- *...
rewrite (Rplus_comm (thetat intr t))...
repeat rewrite <- Rplus_assoc...
rewrite Rplus_opp_r...
rewrite Rplus_0_l...
Qed.

(* Specifications: *)
Lemma yp0 : yp 0 = (l * sin (thetat intr 0 + beta))%R.
unfold yp in |- *; rewrite xe_0; rewrite ye_0;
rewrite <- (Rplus_comm (xi intr 0)); unfold Rminus in |- *;
repeat rewrite Ropp_plus_distr; repeat rewrite <- Rplus_assoc;
repeat rewrite Rplus_opp_r; repeat rewrite Rplus_0_l;
rewrite sin_plus; ring.