
theorem
  for N being Lawson complete TopLattice, T being complete LATTICE, A
being Lawson correct TopAugmentation of T st the RelStr of N = the RelStr of T
  holds the TopRelStr of A = the TopRelStr of N
proof
  let N be Lawson complete TopLattice, T be complete LATTICE, A be Lawson
  correct TopAugmentation of T such that
A1: the RelStr of N = the RelStr of T;
A2: omega T = omega N by A1,WAYBEL19:3;
  set S = the Scott correct TopAugmentation of T;
  set l = the lower correct TopAugmentation of T;
A3: the RelStr of l = the RelStr of T by YELLOW_9:def 4;
A4: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  (the topology of S) \/ (the topology of l) c= bool the carrier of N
  proof
    let a be object;
    assume a in (the topology of S) \/ (the topology of l);
    then a in the topology of S or a in the topology of l by XBOOLE_0:def 3;
    hence thesis by A1,A4,A3;
  end;
  then reconsider
  X = (the topology of S) \/ (the topology of l) as Subset-Family
  of N;
  reconsider X as Subset-Family of N;
A5: the topology of l = omega T by WAYBEL19:def 2;
  (sigma N) \/ (omega N) is prebasis of N by WAYBEL19:def 3;
  then
A6: (sigma T) \/ (omega N) is prebasis of N by A1,YELLOW_9:52;
A7: the topology of S = sigma T by YELLOW_9:51;
  the carrier of N = (the carrier of S) \/ (the carrier of l) by A1,A4,A3;
  then N is Refinement of S, l by A2,A6,A5,A7,YELLOW_9:def 6;
  then
A8: the topology of N = UniCl FinMeetCl X by YELLOW_9:56
    .= lambda T by A1,A5,A7,WAYBEL19:33
    .= the topology of A by WAYBEL19:def 4;
  the RelStr of A = the RelStr of N by A1,YELLOW_9:def 4;
  hence thesis by A8;
end;
