  We consider interacting, charged spins on a torus described by a gapped
Hamiltonian with a unique groundstate and conserved local charge. Using
quasi-adiabatic evolution of the groundstate around a flux-torus, we prove,
without any averaging assumption, that the Hall conductance of the groundstate
is quantized in integer multiples of e^2/h, up to exponentially small
corrections in the linear size of the system. In addition, we discuss
extensions to the fractional quantization case under an additional topological
order assumption on the degenerate groundstate subspace.
