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 Th38: :: Theorem 1.14 (4) implies (1) p. 107
  (for x holds x = "\/" ({inf X : x in X & X in sigma L}, L))
  implies L is continuous
proof
  assume
A1: for x being Element of L holds x = "\/" ({inf V where V is Subset of
L: x in V & V in sigma L}, L);
  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 VV = {inf V where V is Subset of L : x in V & V in sigma L};
A2: sup waybelow x <= x by Th9;
A3: VV c= waybelow x
  proof
    let d be object;
    assume d in VV;
    then consider V being Subset of L such that
A4: inf V = d and
A5: x in V and
A6: V in sigma L;
    V is open by A6,Th24;
    then inf V << x by A5,Th26;
    hence thesis by A4;
  end;
  ex_sup_of VV, L & ex_sup_of waybelow x, L by YELLOW_0:17;
  then
A7: "\/" (VV, L) <= sup waybelow x by A3,YELLOW_0:34;
  x = "\/" (VV, L) by A1;
  hence thesis by A7,A2,ORDERS_2:2;
end;
