  We recall an extension of Kirby's Calculus on non-simply connected
3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke],
which involves only local moves. We give a short combinatorial proof that the
two calculi are equivalent, and thus describe the same classes of 3-manifolds.
This makes the proofs for the validity of surgery calculi in [FR] and [Ke]
interchangeable.
