intros; apply sqrt_inj;
[ apply Rle_0_sqr
| apply Rplus_le_le_0_compat; apply Rle_0_sqr
| rewrite sqrt_Rsqr;
[ unfold u0 in |- *; rewrite Rsqr_sqrt;
[ unfold u in |- *; reflexivity
| apply Rplus_le_le_0_compat; apply Rle_0_sqr ]
| apply (u_pos eps t) ] ].
Qed.

(* Specifications: *)
(**********)
Lemma u0_u_epsilon : forall (eps : posreal) (t : R), (u0 t <= u eps t)%R.
intros; apply Rsqr_incr_0;
[ unfold u0, u in |- *; repeat rewrite Rsqr_sqrt;
[ rewrite <- (Rplus_0_r (Rsqr (xP t) + Rsqr (yP t)));
repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l;
rewrite Rplus_0_l; apply Rle_0_sqr
| repeat apply Rplus_le_le_0_compat; apply Rle_0_sqr
| apply Rplus_le_le_0_compat; apply Rle_0_sqr ]
| apply u0_pos
| apply u_pos ].