Anomaly and renormalization procedure choice

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I've heard an argument, which provides independence of quantum breaking of naive classical symmetry - so-called anomaly - on regularization choice: If we get non-local (here this means non-polynomial) symmetry breaking term in one regularization, it will appear in other ones, since it can't be removed by adding local counterterm. I don't like this argument. It is insufficiently strict, or it is not finished. Could you please comment this argument and provide alternative (or finish it)?

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