  Let $a_{1},...,a_{n}, b_{1},...,b_{n}$ be random variables in some
(non-commutative) probability space, such that $\{a_{1}, ..., a_{n} \}$ is free
from $\{b_{1}, ..., b_{n} \}$. We show how the joint distribution of the
$n$-tuple $(a_{1} b_{1}, ..., a_{n} b_{n})$ can be described in terms of the
joint distributions of $(a_{1}, ..., a_{n})$ and $(b_{1}, >..., b_{n})$, by
using the combinatorics of the $n$-dimensional $R$-transform. We point out a
few applications that can be easily derived from our result, concerning the
left-and-right translation with a semicircular element (see Sections 1.6-1.10)
and the compression with a projection (see Sections 1.11-1.14) of an $n$-tuple
of non-commutative random variables. A different approach to two of these
applications is presented by Dan Voiculescu in an Appendix to the paper.
