intros; cut (u eps t <= vi * t + eps)%R.
intro H0; generalize (u_pos eps t); intro H1;
generalize
(Rsqr_incr_1 (u eps t) (vi * t + eps) H0 H1
(Rle_trans 0 (u eps t) (vi * t + eps) H1 H0));
intro H2; rewrite Rsqr_plus in H2; rewrite Rsqr_u_epsilon in H2;
rewrite Rsqr_mult in H2; rewrite <- (Rplus_comm (2 * (vi * t) * eps)) in H2;
generalize
(Rplus_le_compat_r (- Rsqr eps) (Rsqr (u0 t) + Rsqr eps)
(2 * (vi * t) * eps + (Rsqr vi * Rsqr t + Rsqr eps)) H2);
unfold Rminus in |- *; repeat rewrite Rplus_assoc;
repeat rewrite Rplus_opp_r; repeat rewrite Rplus_0_r;
intro H3;
replace (Rsqr vi * Rsqr t + 2 * eps * vi * t)%R with
(2 * (vi * t) * eps + Rsqr vi * Rsqr t)%R.
assumption.
ring.
apply Rplus_le_reg_l with (- eps)%R; rewrite Rplus_comm;
replace (- eps + (vi * t + eps))%R with (vi * t)%R.
generalize (u_epsilon_le_vt eps t H); unfold Rminus in |- *; intro;
assumption.
ring.
Qed.

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