
theorem Th40:
  for N being meet-continuous Lawson complete TopLattice holds N
  is algebraic iff N is with_open_semilattices & InclPoset sigma N is algebraic
proof
  let N be meet-continuous Lawson complete TopLattice;
  set S = the Scott TopAugmentation of N;
A1: the RelStr of S = the RelStr of N by YELLOW_9:def 4;
  hereby
    assume
A2: N is algebraic;
    then reconsider M = N as algebraic LATTICE;
    M is continuous;
    hence N is with_open_semilattices;
    S is algebraic by A1,A2,WAYBEL_8:17;
    then
    ex K being Basis of S st K = {uparrow x where x is Element of S: x in
    the carrier of CompactSublatt S} by WAYBEL14:42;
    then InclPoset sigma S is algebraic by WAYBEL14:43;
    hence InclPoset sigma N is algebraic by A1,YELLOW_9:52;
  end;
  assume that
A3: N is with_open_semilattices and
A4: InclPoset sigma N is algebraic;
  reconsider T = InclPoset sigma N as algebraic LATTICE by A4;
  T is continuous;
  then N is continuous by A3,Th35;
  then
  for x being Element of S ex K being Basis of x st for Y being Subset of
  S st Y in K holds Y is open filtered by WAYBEL14:35;
  then
A5: for V being Element of InclPoset sigma S ex VV being Subset of
InclPoset sigma S st V = sup VV & for W being Element of InclPoset sigma S st W
  in VV holds W is co-prime by WAYBEL14:39;
  InclPoset sigma S is algebraic by A1,A4,YELLOW_9:52;
  then ex K being Basis of S st K = {uparrow x where x is Element of S: x in
  the carrier of CompactSublatt S} by A5,WAYBEL14:44;
  then S is algebraic by WAYBEL14:45;
  hence thesis by A1,WAYBEL_8:17;
end;
