Section ZL_preorder.
Variable T:Type.
Variable R:relation T.
Hypothesis Rpreord: preorder 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 premaximal (x:T) : Prop :=
forall y:T, R x y -> R y x.
Local Definition Requiv (x y:T) := R x y /\ R y x.
Local Lemma equivalence_Requiv : equivalence Requiv.

Lemma ZornsLemmaForPreorders: exists x:T, premaximal x.