Definition chains_sup (F:Ensemble chains) (P:chain chains_ord F) :=
let U := chains_sup_def F P in
exist (fun U:chains =>
(forall S:chains, In F S -> chains_ord S U) /\
(forall T:chains, (forall S:chains, In F S -> chains_ord S T) ->
chains_ord U T))
(chains_sup_def F P) (chains_sup_correct F P).

Theorem ZornsLemma: exists x:T, maximal R x.
End ZL.