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 :: Theorem 1.14 (2) implies (1) p. 107
  (for X st X in sigma L holds X = union {wayabove x : x in X}) implies
  L is continuous
proof
  assume
A1: for X being Subset of L st X in sigma L holds X = union {wayabove x
  where x is Element of L : x in X};
  thus for x being Element of L holds waybelow x is non empty directed;
  thus L is up-complete;
  let x be Element of L;
  set y = sup waybelow x, X = (downarrow y)`;
  assume
A2: x <> sup waybelow x;
A3: y <= x by Th9;
  now
    assume x in downarrow y;
    then x <= y by WAYBEL_0:17;
    hence contradiction by A3,A2,ORDERS_2:2;
  end;
  then
A4: x in X by XBOOLE_0:def 5;
  set Z = {wayabove z where z is Element of L : z in X};
A5: y is_>=_than waybelow x by YELLOW_0:32;
  X is open by WAYBEL11:12;
  then X in sigma L by Th24;
  then X = union Z by A1;
  then consider Y being set such that
A6: x in Y and
A7: Y in Z by A4,TARSKI:def 4;
  consider z being Element of L such that
A8: Y = wayabove z and
A9: z in X by A7;
  z << x by A6,A8,WAYBEL_3:8;
  then z in waybelow x;
  then z <= y by A5,LATTICE3:def 9;
  then z in downarrow y by WAYBEL_0:17;
  hence contradiction by A9,XBOOLE_0:def 5;
end;
