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We suggest a precursor to phenomena of loss of transient stability in multi-machine power systems. This precursor is based on discovery of [Y. Susuki, I. Mezić , and T. Hikihara, J. Nonlinear Sci., vol. 21, no. 3, pp. 403-439, June 2011], an emergent transmission path of energy from many oscillatory modes to one oscillatory mode that represents an instability phenomenon of interest. The pathway from high frequency modes to the lowest frequency mode is called the coherent swing instability (CSI). The modes are extracted from sensor data or data provided by simulation outputs of power system oscillations by using the Koopman mode analysis that is based on a fully nonlinear spectral theory and represents an extension of linear oscillatory mode analysis. The CSI transmission path is identified by computation of the so-called action transfer operator that is derived by refining a mathematical model of transient stability using the Koopman mode analysis. This provides a new technique for monitoring the loss of transient stability by a combination of practical data, mathematical modeling, and computation.