If a R[X] is a representative of an equivalence class modulo d, then division with remainder leads to an equality a = qd + r with deg (r) < n. Rewriting the equality as a - r = qd shows that a and r are congruent modulo d. Hence r is a representative of degree less than n of the class of a.

Suppose both a and b are representatives of degree less than n of the same class modulo d. Then a and b differ by a multiple of d. Hence a - b = q'd for some polynomial q'. Since the degrees of both a and b are less than the degree of d, the degree of the left-hand side is less than n. But the degree of the right-hand side can only be less than n if q' is the zero polynomial. In particular, a = b.