intros.
generalize
(IAF_var (fun t : R => (vi * t)%R) (fun t : R => u eps t) 0 t vit_derivable
(d_u eps) H); intros.
cut (u eps 0 = eps).
intro H1.
rewrite <- H1.
elim H0; intros.
rewrite Rmult_0_r in H2; unfold Rminus in H2; rewrite Ropp_0 in H2;
rewrite Rplus_0_r in H2; left; assumption.
rewrite Rmult_0_r in H2; unfold Rminus in H2; rewrite Ropp_0 in H2;
rewrite Rplus_0_r in H2; right; assumption.
rewrite (derive_vit c); apply (u_epsilon_le_v eps c).
unfold u in |- *; unfold xP in |- *; unfold yP in |- *; unfold Rminus in |- *;
repeat rewrite Rplus_opp_r; rewrite Rsqr_0; repeat rewrite Rplus_0_l;
apply (sqrt_Rsqr eps (Rlt_le 0 eps (cond_pos eps))).
Qed.

(**********)
Lemma Rsqr_u :
forall (eps : posreal) (t : R),
(0 <= t)%R -> (Rsqr (u0 t) <= Rsqr vi * Rsqr t + 2 * eps * vi * t)%R.