Proof using.
intros.
pose proof max_id_for_client_default_or_entry c l x.
pose proof max_id_for_client_default_or_entry c l' x.
intuition; break_exists; intuition; repeat find_rewrite.
- eapply le_trans; [|eapply Nat.eq_le_incl].
apply max_id_for_client_default_ge_default.
eauto.
- find_apply_hyp_hyp.
eapply le_trans.
+ apply max_id_for_client_default_is_max; eauto.
+ eauto using Nat.eq_le_incl.
- find_apply_hyp_hyp.
eapply le_trans.
+ apply max_id_for_client_default_is_max; eauto.
+ eauto using Nat.eq_le_incl.
Qed.

(* Specifications: *)
Lemma max_id_for_client_default_ext :
forall l l' x c,
(forall e, In e l -> In e l') ->
(forall e, In e l' -> In e l) ->
max_id_for_client_default x c l = max_id_for_client_default x c l'.