
theorem Th14:
  for N being meet-continuous Lawson complete TopLattice, S
  being Scott TopAugmentation of N, A being Subset of N st A in lambda N holds
  uparrow A in sigma S
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, A be Subset of N;
  assume A in lambda N;
  then
A1: A is open by Th12;
A2: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then reconsider Y = A as Subset of S;
A3: S is meet-continuous by A2,YELLOW12:14;
  reconsider UA = uparrow A as Subset of S by A2;
A4: uparrow A = uparrow Y by A2,WAYBEL_0:13;
  Y is property(S) by A1,A2,WAYBEL19:36,YELLOW12:19;
  then UA is open by A3,A4,WAYBEL11:15;
  then uparrow A in the topology of S;
  hence thesis by WAYBEL14:23;
end;
