Same exponents in two nonlinear function deriving the same result?

I have two functions:

$f\left ( X_n \right ) =B*\sum_{n}^{N}X_n^A $

and

$g\left ( X_n \right ) =\sum_{n}^{N}(Z_n*X_n^B) $

with $A$ and $B$ as the exponents of $X_n$.

It alway must hold that $f\left ( X_n \right ) = g\left ( X_n \right )$ for all possible input combinations when changing $X_n$ and keeping all other inputs constant.

I am pretty sure that for this condition to always hold, it must be that A = B. Because only those parameters drive the nonlinearity of those functions and any deviation would not allow for the condition $f\left ( X_n \right ) = g\left ( X_n \right )$, no matter what values the other variables $B$, $Z_n$ and $N$ take.

Can this intuitive result be proven mathematically in easy terms?

Thanks for your support!

Best