  We have recently proposed a new lattice SUSY formulation which has exact
lattice supersymmetry for Wess-Zumino models in one and two dimensions for all
N=2 supercharges. This formulation is non-local in the coordinate space but the
difference operator satisfies the Leibniz rule on the newly defined star
product. Here we show that this lattice supersymmetry is kept exact at the
quantum level by investigating Ward-Takahashi identities up to two loop level.
