
theorem Th17:
  for N being meet-continuous Lawson complete TopLattice for S
being Scott TopAugmentation of N for X being upper Subset of N, Y being Subset
  of S st X = Y holds Int X = Int Y
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, X be upper 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 K = uparrow Int X as Subset of S;
  reconsider sX = Int X as Subset of S by A2;
A3: K c= Y
  proof
    let a be object;
A4: Int X c= X by TOPS_1:16;
    assume
A5: a in K;
    then reconsider x = a as Element of N;
A6: X c= uparrow X by WAYBEL_0:16;
    uparrow X c= X by WAYBEL_0:24;
    then
A7: uparrow X = X by A6;
    ex y being Element of N st y <= x & y in Int X by A5,WAYBEL_0:def 16;
    hence thesis by A7,A1,A4,WAYBEL_0:def 16;
  end;
A8: Int X c= uparrow Int X by WAYBEL_0:16;
  uparrow sX is open by Th15;
  then K is open by A2,WAYBEL_0:13;
  then uparrow Int X c= Int Y by A3,TOPS_1:24;
  hence Int X c= Int Y by A8;
  reconsider A = Int Y as Subset of N by A2;
  N is Lawson correct TopAugmentation of S by A2,YELLOW_9:def 4;
  then A is open by WAYBEL19:37;
  hence thesis by A1,TOPS_1:16,24;
end;
