reserve X for non empty TopSpace;

theorem Th2:
  for M being Subset of X holds M is maximal_T_0 implies M is dense
proof
  let M be Subset of X;
  assume
A1: M is maximal_T_0;
  then MaxADSet(M) = [#]X;
  then
A2: Cl MaxADSet(M) = MaxADSet(M) by PRE_TOPC:22;
  MaxADSet(M) = the carrier of X by A1;
  then Cl M = the carrier of X by A2,TEX_4:62;
  hence thesis by TOPS_3:def 2;
end;
