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

theorem
  for X being non empty TopSpace holds (for A being Subset of X holds (A
  <> the carrier of X implies A is boundary)) implies X is anti-discrete
proof
  let X be non empty TopSpace;
  assume
A1: for A being Subset of X holds (A <> the carrier of X implies A is
  boundary);
  now
    let A be Subset of X;
    reconsider B = A as Subset of X;
    assume A <> the carrier of X;
    then B is boundary by A1;
    hence Int A = {};
  end;
  hence thesis by Th12;
end;
