Title: Jarzynski equality for time-averaged work

Abstract: There is evidence that taking the time average of the work performed by a thermally isolated system "transforms" the adiabatic process into an isothermal one. We add here one more fact to this case by showing that the time-averaged difference of Helmholtz free energy is equal to the time-averaged quasistatic work of the system. Therefore, a Jarzynski equality relating the time-averaged work and time-averaged quasistatic work is established. As an immediate consequence, the expected relation $\langle W\rangle\ge W_{\rm qs}$ is demonstrated. Numerical evidence for the equality is also presented for the classical harmonic oscillator with a driven linear equilibrium position parameter.