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.