
theorem
  for N being meet-continuous Lawson complete TopLattice for S being
  Scott TopAugmentation of N for X being lower Subset of N, Y being Subset of S
  st X = Y holds Cl X = Cl Y
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, X be lower Subset of N, Y be Subset of S such that
A1: X = Y;
A2: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then reconsider A = Cl Y as Subset of N;
  (Cl X)` = (Cl X``)` .= Int X` by TOPS_1:def 1
    .= Int Y` by A1,A2,Th17
    .= (Cl Y``)` by TOPS_1:def 1
    .= A` by A2;
  hence thesis by TOPS_1:1;
end;
