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

theorem ::3.2, p.68
  for L be up-complete LATTICE, X be upper Subset of L holds X is Open
  iff X = union {wayabove x where x is Element of L : x in X}
proof
  let L be up-complete LATTICE, X be upper Subset of L;
  hereby
    assume
A1: X is Open;
A2: X c= union {wayabove x where x is Element of L : x in X}
    proof
      let z be object;
      assume
A3:   z in X;
      then reconsider x1 = z as Element of X;
      reconsider x1 as Element of L by A3;
      consider y be Element of L such that
A4:   y in X and
A5:   y << x1 by A1,A3;
      x1 in {y1 where y1 is Element of L: y1 >> y} by A5;
      then
A6:   x1 in wayabove y by WAYBEL_3:def 4;
      wayabove y in {wayabove x where x is Element of L : x in X} by A4;
      hence thesis by A6,TARSKI:def 4;
    end;
    union {wayabove x where x is Element of L : x in X} c= X
    proof
      let z be object;
      assume z in union {wayabove x where x is Element of L : x in X};
      then consider Y be set such that
A7:   z in Y and
A8:   Y in {wayabove x where x is Element of L : x in X} by TARSKI:def 4;
      consider x be Element of L such that
A9:   Y = wayabove x and
A10:  x in X by A8;
      z in {y where y is Element of L: y >> x} by A7,A9,WAYBEL_3:def 4;
      then consider z1 be Element of L such that
A11:  z1 = z and
A12:  z1 >> x;
      x <= z1 by A12,WAYBEL_3:1;
      hence thesis by A10,A11,WAYBEL_0:def 20;
    end;
    hence X = union {wayabove x where x is Element of L : x in X} by A2;
  end;
  assume
A13: X = union {wayabove x where x is Element of L : x in X};
  now
    let x1 be Element of L;
    assume x1 in X;
    then consider Y be set such that
A14: x1 in Y and
A15: Y in {wayabove x where x is Element of L : x in X} by A13,TARSKI:def 4;
    consider x be Element of L such that
A16: Y = wayabove x and
A17: x in X by A15;
    x1 in {y where y is Element of L: y >> x} by A14,A16,WAYBEL_3:def 4;
    then ex z1 be Element of L st z1 = x1 & z1 >> x;
    hence ex x be Element of L st x in X & x << x1 by A17;
  end;
  hence thesis;
end;
