
theorem Th15:
  for N being meet-continuous Lawson complete TopLattice for S
  being Scott TopAugmentation of N for A being Subset of N, J being Subset of S
  st A = J holds A is open implies uparrow J is open
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, A be Subset of N, J be Subset of S such that
A1: A = J;
  assume A is open;
  then A in lambda N by Th12;
  then
A2: uparrow A in sigma S by Th14;
  the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then uparrow J in sigma S by A2,A1,WAYBEL_0:13;
  hence thesis by WAYBEL14:24;
end;
