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 discrete iff for A being Subset of X holds A <> the carrier of X
  implies A is not dense
proof
  hereby
    assume
A1: X is discrete;
    assume not for A being Subset of X holds A <> the carrier of X implies A
    is not dense;
    then consider A being Subset of X such that
A2: A <> the carrier of X and
A3: A is dense;
    now
      reconsider B = A` as non empty Subset of X by A2,TOPS_3:2;
      take B;
      thus B is boundary by A3,TOPS_3:18;
    end;
    hence contradiction by A1;
  end;
  assume
A4: for C being Subset of X holds C <> the carrier of X implies C is not dense;
  assume X is non discrete;
  then consider A being non empty Subset of X such that
A5: A is boundary;
  now
    take B = A`;
    thus B <> the carrier of X & B is dense by A5,TOPS_1:def 4,TOPS_3:1;
  end;
  hence contradiction by A4;
end;
