I need some help with Uspensky's proof on the sum of three squares found in "Elementary Number Theory" by Uspensky and Heaslet. I have gotten to the end of the proof and do not understand a few statements found at the end. I was wondering if anyone was familiar with the proof. In it he defines T(n) to be the total solutions to
With a+b  \equiv 0 \pmod 4 and a,b,c all odd.
Let N_3(n) be the number of representations of n as the sum of three squares.
He shows 2[T(4d-1^2)+T(4d-3^2)+T(4d-5^2)+\ldots ]=N_3(4d-1^2)+N_3(4d-3^2)+N_3(4d-5^2)+\ldots
Which I understand where that comes from. What I am confused on, is he claims that since this identity is true for all odd values of m that we must have
Why must we have that? Any help would be greatly appreciated.