
theorem
  for N being meet-continuous Hausdorff Lawson complete TopLattice, x
being Element of N holds x = "\/"({inf V where V is Subset of N: x in V & V in
  lambda N},N)
proof
  let N be meet-continuous Hausdorff Lawson complete TopLattice, x be
  Element of N;
  set S = the Scott complete TopAugmentation of N;
A1: InclPoset sigma S is continuous by WAYBEL14:36;
A2: the RelStr of S = the RelStr of N by YELLOW_9:def 4;
  then reconsider y = x as Element of S;
  for y being Element of S ex J being Basis of y st for X being Subset of
  S st X in J holds X is open filtered by WAYBEL14:35;
  hence x = "\/" ({inf X where X is Subset of S: y in X & X in sigma S}, S) by
A1,WAYBEL14:37
    .= "\/" ({inf X where X is Subset of S: x in X & X in sigma S}, N) by A2,
YELLOW_0:17,26
    .= "\/" ({inf V where V is Subset of N: x in V & V in lambda N}, N) by Th34
;
end;
