reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;

theorem
  for a being Element of R holds a is_associated_to 0.R implies a = 0.R
proof
  let A be Element of R;
  assume A is_associated_to 0.R;
  then 0.R divides A;
  then ex D being Element of R st 0.R * D = A;
  hence thesis;
end;
