reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem
  X is non almost_discrete iff ex A being Subset of X st A <> the
  carrier of X & A is dense & A is open
proof
  thus X is non almost_discrete implies ex A being Subset of X st A <> the
  carrier of X & A is dense & A is open
  proof
    assume X is non almost_discrete;
    then consider A being non empty Subset of X such that
A1: A is boundary and
A2: A is closed by Th33;
    take A`;
    thus thesis by A1,A2,TOPS_3:1;
  end;
  given A being Subset of X such that
A3: A <> the carrier of X and
A4: A is dense and
A5: A is open;
  now
    reconsider B = A` as non empty Subset of X by A3,TOPS_3:2;
    take B;
    thus B is boundary & B is closed by A4,A5,TOPS_3:18;
  end;
  hence thesis;
end;
