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

theorem Th7:
  for C being Subset of X modified_with_respect_to D` st C c= D
  holds D is boundary implies C is nowhere_dense
proof
  let C be Subset of X modified_with_respect_to D`;
  assume
A1: C c= D;
  reconsider E = D as Subset of X modified_with_respect_to D` by TMAP_1:93;
  assume
A2: D is boundary;
  then
A3: E is closed by Th6;
  E is boundary by A2,Th6;
  hence thesis by A1,A3,TOPS_3:26;
end;
