intros; replace (hx t) with (hx t - hx 0)%R.
apply (IAF_var xp hx 0 t xp_derivable hx_derivable H); intros; rewrite Dhx;
elim H1; intros;
apply
(Dx0_PI c H2 (Rle_trans (rho_vi * c) (rho_vi * t) PI (rho_t_le c t H3) H0)).
unfold hx in |- *; rewrite Rmult_0_r; rewrite sin_0; unfold Rminus in |- *;
rewrite Rmult_0_r; rewrite Ropp_0; apply Rplus_0_r.
Qed.

Lemma r_sin_pos :
forall t : R,
(0 <= t)%R -> (rho_vi * t <= PI)%R -> (0 <= r_vi * sin (rho_vi * t))%R.