3 corrected definition of proximality

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.

He proved in Theorem 8.8 that in each finite partition of $\mathbb{N}$, $\mathbb{N}=B_1 \cup B_2 \cup \dots \cup B_q$ one of the sets $B_j$ contains a central set. Then he states without proof that the same is true for partitions of central sets and that one can prove this "along the same lines" as Theorem 8.8 (p. 163).

I know that this fact follows immediately from the characterization of central sets by minimal idempotent ultrafilters. However, this is obviously not the proof that Furstenberg had in mind 1981 as this characterization was not available then. Thus my question:

How can one prove in the fashion of Theorem 8.8, i.e. using only tools recurrence in dynamical systems, that for each finite partition of a central set $S$, $S=B_1 \cup B_2 \cup \dots \cup B_q$ one of the $B_j$ contains a set that is again central?

For the sake of reference I include the necessary definitions and the proof of Theorem 8.8 below.