  We show that the braided tensor product algebra $A_1\underline{\otimes}A_2$
of two module algebras $A_1, A_2$ of a quasitriangular Hopf algebra $H$ is
equal to the ordinary tensor product algebra of $A_1$ with a subalgebra of
$A_1\underline{\otimes}A_2$ isomorphic to $A_2$, provided there exists a
realization of $H$ within $A_1$. In other words, under this assumption we
construct a transformation of generators which `decouples' $A_1, A_2$ (i.e.
makes them commuting). We apply the theorem to the braided tensor product
algebras of two or more quantum group covariant quantum spaces, deformed
Heisenberg algebras and q-deformed fuzzy spheres.
