  We give a new, and hopefully more easily understandable, structural proof of
the decomposition of a $k$-valued transducer into $k$ unambiguous functional
ones, a result established by A. Weber in 1996. Our construction is based on a
lexicographic ordering of computations of automata and on two coverings that
can be build by means of this ordering. The complexity of the construction,
measured as the number of states of the transducers involved in the
decomposition, improves the original one by one exponential. Moreover, this
method allows further generalisation that solves the problem of decomposition
of rational relations with bounded length-degree, which was left open in
Weber's paper.
