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

theorem
  for X being anti-discrete non empty TopSpace for A being Subset of X
  holds (A <> {} implies A is dense) & (A <> the carrier of X implies A is
  boundary)
proof
  let X be anti-discrete non empty TopSpace;
  let A be Subset of X;
  thus A <> {} implies A is dense
  proof
    assume A <> {};
    then Cl A = the carrier of X by Th9;
    hence thesis by TOPS_3:def 2;
  end;
  thus A <> the carrier of X implies A is boundary
  proof
    assume A <> the carrier of X;
    then Int A = {} by Th10;
    hence thesis by TOPS_1:48;
  end;
end;
