 In the limit where the coupling constants we have added go to zero, one gets back to the original theory, plus the fermions we have added; the latter remain good degrees of freedom at every energy scale, as they are free fermions at this limit.The gauge symmetry anomaly can be computed at any energy scale, and must always be zero, so that the theory is consistent.One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.