A proof of the Cramér-Rao inequality for prediction is presented under conditions of L2-differentiability of the family of distributions of the model. The assumptions and the proof differ from those of Miyata (2001) who also proved this inequality under L2-differentiability conditions. It is also proved that if an efficient predictor (i.e. which risk attains the bound) exists then the family of distributions is of a special form which can be seen as an extension of the notion of exponential family. This result is also proved under L2-differentiability conditions.