
theorem
  for N being meet-continuous Lawson complete TopLattice holds N is
continuous iff for 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 Lawson complete TopLattice;
  set S = the complete Scott TopAugmentation of N;
A1: the RelStr of S = the RelStr of N by YELLOW_9:def 4;
  hereby
    assume
A2: N is continuous;
    then
A3: for x being Element of S ex J being Basis of x st for Y being Subset
    of S st Y in J holds Y is open filtered by WAYBEL14:35;
    let x be Element of N;
    InclPoset sigma S is continuous by A2,WAYBEL14:36;
    hence x = "\/" ({inf X where X is Subset of S: x in X & X in sigma S}, S )
    by A3,A1,WAYBEL14:37
      .= "\/" ({inf X where X is Subset of S: x in X & X in sigma S}, N) by A1,
YELLOW_0:17,26
      .= "\/"({inf V where V is Subset of N: x in V & V in lambda N},N) by Th34
;
  end;
  assume
A4: for x being Element of N holds x = "\/"({inf V where V is Subset of
N: x in V & V in lambda N},N);
  now
    let x be Element of S;
    thus x = "\/"({inf V where V is Subset of N: x in V & V in lambda N},N) by
A1,A4
      .= "\/"({inf V where V is Subset of S: x in V & V in sigma S},N) by A1
,Th34
      .= "\/"({inf V where V is Subset of S: x in V & V in sigma S},S) by A1,
YELLOW_0:17,26;
  end;
  then S is continuous by WAYBEL14:38;
  hence thesis by A1,YELLOW12:15;
end;
