  We introduce an addition law for the usual quantum matrices $A(R)$ by means
of a coaddition $\underline{\Delta} t=t\otimes 1+1\otimes t$. It supplements
the usual comultiplication $\Delta t=t\otimes t$ and together they obey a
codistributivity condition. The coaddition does not form a usual Hopf algebra
but a braided one. The same remarks apply for rectangular $m\times n$ quantum
matrices. As an application, we construct left-invariant vector fields on
$A(R)$ and other quantum spaces. They close in the form of a braided Lie
algebra. As another application, the wave-functions in the lattice
approximation of Kac-Moody algebras and other lattice fields can be added and
functionally differentiated.
