If a Mealy machine decomposes into prime Mealy machines, then so does its map lifting (Lemma A.2.4 of Transducers). Since map lifting commutes with composition, it suffices to lift a single prime. A flip-flop is lifted by resetting its state at every separator. A reversible machine needs more care, because resetting would break reversibility: the book computes, by a reversible machine, the state transformation $\gamma_w$ of every prefix in the variant where the separator does nothing, stores the value at the last separator by a flip-flop delay machine, and recovers the state transformation of the current block by the cancellation law $\delta_w = \gamma_v^{-1} \cdot \gamma_{vw}$, all of which is a one-state machine's work.