Summary: We extend the well-known P. Lévy theorem on the distributional identity \((M_t-B_t,M_t) \simeq(|B_t|,\;L(B)_t)\), where \((B_t)\) is a standard Brownian motion and \((M_t)= (\sup_{0\leq s\leq t}B_s)\) to the case of Brownian motion with drift \(\lambda\). Processes of the type \(dX_t^\lambda= -\lambda \text{sgn}(X_t^\lambda) dt+ dB_t\) appear naturally in the generalization.