reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem
  for L be up-complete LATTICE, X be upper Subset of L holds X is Open
  iff for x be Element of L st x in X holds waybelow x meets X
proof
  let L be up-complete LATTICE, X be upper Subset of L;
  hereby
    assume
A1: X is Open;
    thus for x be Element of L st x in X holds waybelow x meets X
    proof
      let x be Element of L;
      assume x in X;
      then consider y be Element of L such that
A2:   y in X and
A3:   y << x by A1;
      y in {y1 where y1 is Element of L: y1 << x} by A3;
      then y in waybelow x by WAYBEL_3:def 3;
      hence thesis by A2,XBOOLE_0:3;
    end;
  end;
  assume
A4: for x be Element of L st x in X holds waybelow x meets X;
  now
    let x1 be Element of L;
    assume x1 in X;
    then (waybelow x1) meets X by A4;
    then consider y being object such that
A5: y in (waybelow x1) and
A6: y in X by XBOOLE_0:3;
    waybelow x1 = {y1 where y1 is Element of L: y1 << x1} by WAYBEL_3:def 3;
    then ex z be Element of L st z = y & z << x1 by A5;
    hence ex z be Element of L st z in X & z << x1 by A6;
  end;
  hence thesis;
end;
