@MISC{RELSET_2.MIZ,
  AUTHOR = {Retel, Krzysztof},
  TITLE = {Properties of First and Second Order Cutting of
           Binary Relations},
  ACKNOWLEDGEMENT = {The author would like to express his gratitude to Professor
    Grzegorz Bancerek for many helpful conversations and support during the preparation of this paper.},
  SECTION1 = {Preliminaries},
  SECTION2 = {The first order cutting of binary relation of two sets A,B under subset of the set A},
  SECTION3 = {The second order cutting of binary relation of two sets A,B under subset of the set A},
  DAY = {25},
  MONTH = {April},
  YEAR = {2005},
  ADDRESS1 = {Heriot-Watt University\\School of Mathematical and Computer Sciences\\Riccarton, Edinburgh EH14 4AS},
  SUMMARY = {This paper introduces some notions concerning binary relations according to \cite{Riguet}.
It is also an attempt to complement the knowledge contained in Mizar Mathematical Library regarding binary relations.
We define here an image and inverse image of element of set A under binary relation of two sets A,B as image and 
inverse image of singleton of the element under this relation, respectively. 
Next, we define "The First Order Cutting Relation of two sets A,B under a subset of the set A" 
as the union of images of elements of this subset under the relation.
We also define "The Second Order Cutting Subset of the Cartesian Product of two sets A,B under a subset of the set A"
as an intersection of images of elements of this subset under the subset of the cartesian product.
The paper also defines first and second projection of binary relations.
The main goal of the article is to prove properties and collocations of introduced definitions in this paper.
The numbers written in parenthesis after the label of theorems correspond to the numbers of expressions contained in 
the original article.}}


@ARTICLE{Riguet,
       AUTHOR = {Riguet, Jacques},
        TITLE = {Relations binaires, fermetures, correspondances de Galois},
      JOURNAL = {Bulletin de la S.M.F.},
         YEAR = {1948},
       VOLUME = {76},
        PAGES = {114--155},
         NOTE = {On WWW: {\tt http://www.numdam.org/item?id=BSFM_1948__76__114_0}}}

