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

theorem Th32:
  X is almost_discrete iff for A being Subset of X holds A <> the
  carrier of X implies A is not everywhere_dense
proof
  hereby
    assume
A1: X is almost_discrete;
    assume not for A being Subset of X holds A <> the carrier of X implies A
    is not everywhere_dense;
    then consider A being Subset of X such that
A2: A is everywhere_dense and
A3: A <> the carrier of X;
    now
      reconsider B = A` as non empty Subset of X by A3,TOPS_3:2;
      take B;
      thus B is nowhere_dense by A2,TOPS_3:39;
    end;
    hence contradiction by A1;
  end;
  assume
A4: for A being Subset of X holds A <> the carrier of X implies A is not
  everywhere_dense;
  assume X is non almost_discrete;
  then consider A being non empty Subset of X such that
A5: A is nowhere_dense;
  now
    take B = A`;
    thus B <> the carrier of X & B is everywhere_dense by A5,TOPS_3:1,40;
  end;
  hence contradiction by A4;
end;
