Every rational function is computed by a bimachine: the implication (1) ⇒ (3) of Theorem B.2.3 of Transducers (Eilenberg), through the unambiguous automaton of (2). Given an unambiguous automaton whose accepting runs end with a single empty-input transition, the bimachine's prefix automaton computes the states reachable from an initial state on the prefix, its suffix automaton computes the first letter of the suffix and the states that can reach acceptance on the rest, and the output function reads off the output of the unique transition consuming the letter at the gap — or of the final empty-input transition at the last gap.