The quotient-remainder theorem says not only that there exist quotients and remainders but...

The quotient-remainder theorem says not only that there exist quotients and remainders but also that the quotient and remainder of a division are unique. Prove the uniqueness. That is, prove that if a and d are integers with d > 0 and if q1, r1, q2, and r2 are integers such that

a = dq1 + r1 where 0 ≤ r1<d

and

a = dq2 + r2 where 0<r2<d,

then

q1 = q2 and r1 = r2.