(*  remember half as h. *)
intuition.
unfold inf_limit in *.
intuition.
specialize (H1 (half epsilon)).
specialize (H2 (half epsilon)).
destruct H1.
intuition.
eapply half_nz; eauto.
destruct H2.
intuition.
eapply half_nz; eauto.
destruct (H (max x x0)).
specialize (H1 (max x x0)).
specialize (H2 (max x x0)).
eapply le_trans.
eapply (triangle_inequality _ _ x1).
eapply le_trans.
eapply plus_le_compat.
eapply le_trans.
eapply le_eq.
eapply distance_comm.
eapply H1.
eapply Max.max_lub_l; eauto.
trivial.
eapply H2.
eapply Max.max_lub_r; eauto.
trivial.
eapply le_eq.
eapply half_plus.
Qed.

Theorem limits_eq : forall f a1 a2,
left_total f ->
inf_limit f a1 ->
inf_limit f a2 ->
eq a1 a2.