
theorem Th7:
  for X being set, Y being Subset of BoolePoset X holds Y is upper
  iff for x,y being set st x c= y & y c= X & x in Y holds y 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 upper;
    let x,y be set;
    assume that
A3: x c= y and
A4: y c= X and
A5: x in Y;
    reconsider a = x, b = y as Element of BoolePoset X by A4,A5,Th2;
    a <= b by A3,YELLOW_1:2;
    hence y in Y by A2,A5;
  end;
  assume
A6: for x,y being set st x c= y & y c= X & x in Y holds y in Y;
  let a,b be Element of BoolePoset X;
  assume that
A7: a in Y and
A8: b >= a;
  a c= b by A8,YELLOW_1:2;
  hence thesis by A1,A6,A7;
end;
