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We present a cryptographically sound security proof of the well-known Needham-Schroeder-Lowe public-key protocol for entity authentication. This protocol was previously only proved over unfounded abstractions from cryptography. We show that it is secure against arbitrary active attacks if it is implemented using standard provably secure cryptographic primitives. Nevertheless, our proof does not have to deal with the probabilistic aspects of cryptography and is, hence, in the scope of current automated proof tools. We achieve this by exploiting a recently proposed Dolev-Yao-style cryptographic library with a provably secure cryptographic implementation. Besides establishing the cryptographic security of the Needham-Schroeder-Lowe protocol, our result exemplifies the potential of this cryptographic library and paves the way for the cryptographically sound verification of security protocols by automated proof tools.