@MISC{ORDERS_2.MIZ,
  AUTHOR = {Trybulec, Wojciech A. and Bancerek, Grzegorz},
  TITLE = {Kuratowski - {Z}orn Lemma},
  NOTE = {Supported by RPBP.III-24.C1.},
  DAY = {19},
  MONTH = {September},
  YEAR = {1989},
  ADDRESS1 = {Warsaw University},
  ADDRESS2 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {The goal of this article is to prove Kuratowski - Zorn lemma. We prove it
in a number of forms (theorems and schemes).
We introduce the following notions:
 a relation is a quasi (or partial, or linear) order,
a relation quasi (or partially, or linearly) orders a set,
minimal and maximal element in a relation,
 inferior and superior element of a relation,
a set has lower (or upper) Zorn property w.r.t. a relation.
 We prove basic theorems concerning those notions and theorems
 that relate them to the notions introduced in \cite{ORDERS_1.ABS}.
 At the end of the article we prove some theorems that belong rather
 to \cite{RELAT_1.ABS}, \cite{RELAT_2.ABS} or \cite{WELLORD1.ABS}.}}

