The theorem goes as follows:
Suppose the family of densities $\{f_0(x), .., f_k(0)\}$ all have common support. Then,
(a)
$$T(X) = \Big(\frac{f_1(X)}{f_0(X)}, \frac{f_2(X)}{f_0(X)}, .., \frac{f_k(X)}{f_0(X)}\Big)$$
is minimal sufficient for the family above.
(b) If $\mathcal{F}$ is a family of densities with common support, and
$\quad$ i) $f_i(x) \in \mathcal{F}, i= 0, 1, .., k$
$\quad$ ii) T(x) is sufficient for $\mathcal{F}$
then $T(x)$ is minimal sufficient for $\mathcal{F}$.
I already proved (a), but the textbook (Casella-Berger exercise 6.28) suggests to show that any sufficient statistic of $\mathcal{F}$ is a function of $T(x)$ defined in (a) somehow.
I was thinking of picking a random set of densities from $\mathcal{F}$, assuming that $T(x) = T(y)$, and then showing that some arbitrary sufficient statistic $T'(x) = T'(y)$ for $\mathcal{F}$ is implied. But somehow got stuck.