Proof with trivial.
intro; unfold hx in |- *; reg...
replace (vi intr:R) with (r_vi * rho_vi)%R.
ring.
unfold rho_vi, r_vi in |- *; unfold r in |- *.
field.
assert (H := rho_pos (vi intr)); red in |- *; intro; rewrite H0 in H;
elim (Rlt_irrefl _ H)...
Qed.

Lemma xpt :
forall t : R, (0 <= t)%R -> (rho_vi * t <= PI)%R -> (hx t <= xp t - xp 0)%R.