Proof
To show that congruence modulo d is an equivalence relation, we have to verify that this relation is
reflexive,
symmetric, and
transitive.
This follows from the fact that for every polynomial a we have:
a - a = 0 · d.
If a and b are congruent modulo d,
i.e., if a - b = qd
for some polynomial q, then rewriting this equality
as b - a = -qd shows
that b and a are also congruent modulo
d. Therefore the relation is symmetric.
If a is congruent to b modulo d
and b is congruent to c modulo d,
then there exist polynomials q and p with
a - b = qd
and b - c = pd.
Adding these equalities yields the equality a - c =
(q + p)d.
Hence a and c are congruent modulo d.