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Recently, Vaidyanathan  presented the discrete-time bounded-real lemma and lossless bounded-real lemma. He derived the lemmas employing energy-balance arguments. The present paper provides an alternative proof of these lemmas. The new proof is very similar to that employed previously ,  in connection with the proof of the discretetime version of the MKY lemma. Note: The notations used in the present paper have the same meaning as in .