reserve X for non empty TopSpace,
  D for Subset of X;

theorem Th6:
  for C being Subset of X modified_with_respect_to D` st C = D
  holds D is boundary implies C is boundary & C is closed
proof
  let C be Subset of X modified_with_respect_to D`;
  assume C = D;
  then
A1: C` = D` by TMAP_1:93;
  assume D is boundary;
  then
A2: D` is dense by TOPS_1:def 4;
  then
A3: C` is open by A1,Th4;
  C` is dense by A1,A2,Th4;
  hence thesis by A3,TOPS_1:def 4;
end;
