reserve L for complete Scott TopLattice,
  x for Element of L,
  X, Y for Subset of L,
  V, W for Element of InclPoset sigma L,
  VV for Subset of InclPoset sigma L;

theorem Th33: :: Theorem 1.14 (1) implies (2) p. 107
  L is continuous & X in sigma L implies X = union {wayabove x : x in X}
proof
  assume that
A1: L is continuous and
A2: X in sigma L;
  set WAV = {wayabove x where x is Element of L : x in X};
A3: X is open by A2,Th24;
  now
    let x be object;
    hereby
      assume
A4:   x in X;
      then reconsider x9 = x as Element of L;
      consider q being Element of L such that
A5:   q << x9 & q in X by A1,A3,A4,WAYBEL11:43;
      x9 in wayabove q & wayabove q in WAV by A5;
      hence x in union WAV by TARSKI:def 4;
    end;
    assume x in union WAV;
    then consider Y being set such that
A6: x in Y and
A7: Y in WAV by TARSKI:def 4;
    consider q being Element of L such that
A8: Y = wayabove q and
A9: q in X by A7;
A10: wayabove q c= uparrow q by WAYBEL_3:11;
    X is upper by A3,WAYBEL11:def 4;
    then uparrow q c= X by A9,WAYBEL11:42;
    then Y c= X by A8,A10;
    hence x in X by A6;
  end;
  hence thesis by TARSKI:2;
end;
