we must have n=k and the only possible value of m in the sum is j,
so that the result is trivial. Now assume that 
, and write
m and n in base p. Then
(25)
for each m in the sum in (11), as 
for each i.
Thus, by Lucas' Theorem, the sum in (11) is congruent to
where the sum is over all 
-tuples of integers 
satisfying (25) and not all zero. This
is exactly the sum of the coefficients of
in
,
which equals
(11) then follows from the induction hypothesis as
and 
.