
theorem Th9:
  for X being set, Y being upper Subset of BoolePoset X holds Y is
  filtered iff for x,y being set st x in Y & y in Y holds x /\ y in Y
proof
  let X be set, Y be upper Subset of BoolePoset X;
  hereby
    assume
A1: Y is filtered;
    let x,y be set;
    assume
A2: x in Y & y in Y;
    then reconsider a = x, b = y as Element of BoolePoset X;
    a"/\"b in Y by A1,A2,WAYBEL_0:41;
    hence x /\ y in Y by YELLOW_1:17;
  end;
  assume
A3: for x,y being set st x in Y & y in Y holds x /\ y in Y;
  now
    let a,b be Element of BoolePoset X;
    assume a in Y & b in Y;
    then a /\ b in Y by A3;
    hence a"/\"b in Y by YELLOW_1:17;
  end;
  hence thesis by WAYBEL_0:41;
end;
