let be the set of all odd divisors of and the condition

is obviously equivalent to now define by:

it's easy to see that is well-defined and injective. to prove that is surjective, pick if is even or zero, then let and

if is odd, then let it should be straightforward for you to see that so is a bijection and we're done!

by the way just means that "the proof is complete!" i think it looks better than !