reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;

theorem
  for Y0 being closed non empty SubSpace of X for A being Subset of X, B
  being Subset of Y0 st A = B holds Cl A = Cl B
proof
  let Y0 be closed non empty SubSpace of X;
  reconsider C = the carrier of Y0 as Subset of X by TSEP_1:1;
  let A be Subset of X, B be Subset of Y0;
A1: C is closed by TSEP_1:11;
  assume A = B;
  hence thesis by A1,Th54;
end;
