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

theorem
  for A being Subset of X st A <> the carrier of X & A is dense holds X
  modified_with_respect_to A is non almost_discrete
proof
  let A be Subset of X;
  assume
A1: A <> the carrier of X;
  set Z = X modified_with_respect_to A;
  assume
A2: A is dense;
  now
    reconsider C = A as Subset of Z by TMAP_1:93;
    take C;
    thus C <> the carrier of Z & C is everywhere_dense by A1,A2,Th5,TMAP_1:93;
  end;
  hence thesis by Th32;
end;
