theorem
  for A being non empty Subset of X st A is boundary holds X
  modified_with_respect_to A` is non almost_discrete
proof
  let A be non empty Subset of X;
  set Z = X modified_with_respect_to A`;
  assume
A1: A is boundary;
  now
    reconsider C = A as non empty Subset of Z by TMAP_1:93;
    take C;
    thus C is nowhere_dense by A1,Th7;
  end;
  hence thesis;
end;
