
theorem
  for X being set, Y being Subset of BoolePoset X holds Y is lower iff
  for x,y being set st x c= y & y in Y holds x in Y
proof
  let X be set, Y be Subset of BoolePoset X;
A1: the carrier of BoolePoset X = bool X by Th2;
  hereby
    assume
A2: Y is lower;
    let x,y be set;
    assume that
A3: x c= y and
A4: y in Y;
    reconsider a = x, b = y as Element of BoolePoset X by A1,A3,A4,XBOOLE_1:1;
    a <= b by A3,YELLOW_1:2;
    hence x in Y by A2,A4;
  end;
  assume
A5: for x,y being set st x c= y & y in Y holds x in Y;
  let a,b be Element of BoolePoset X;
  assume that
A6: a in Y and
A7: b <= a;
  b c= a by A7,YELLOW_1:2;
  hence thesis by A5,A6;
end;
