
theorem Th18:
  for C being FormalContext holds (AttributeDerivation(C)).{} =
  the carrier of C
proof
  let C be FormalContext;
  reconsider e = {} as Subset of the carrier' of C by XBOOLE_1:2;
  set O = {o where o is Object of C : for a being Attribute of C st a in e
  holds o is-connected-with a};
A1: for x being object holds x in O implies x in the carrier of C
  proof
    let x be object;
    assume x in O;
    then
    ex x9 being Object of C st x9 = x & for a being Attribute of C st a in
    e holds x9 is-connected-with a;
    hence thesis;
  end;
A2: for x being object holds x in the carrier of C implies x in O
  proof
    let x be object;
    assume x in the carrier of C;
    then reconsider x as Object of C;
    for a being Attribute of C st a in e holds x is-connected-with a;
    hence thesis;
  end;
  (AttributeDerivation(C)).e = {o where o is Object of C : for a being
  Attribute of C st a in e holds o is-connected-with a} by Def3;
  hence thesis by A1,A2,TARSKI:2;
end;
