Section ZL.
(* get rid of the need for a sup function and immediate successors *)
Variable T:Type.
Variable R:relation T.
Hypothesis ord: order R.
Hypothesis ub_of_chain: forall S:Ensemble T, chain R S ->
exists x:T, forall y:T, In S y -> R y x.
Definition chains := {S:Ensemble T | chain R S}.
Definition chains_ord := (fun S1 S2:chains =>
Included (proj1_sig S1) (proj1_sig S2)).

Lemma chains_order: order chains_ord.