Delta integral of certain shift of f is delta integral of f

Theorem

The following formula holds: $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ where $\displaystyle\int$ denotes the delta integral, $\sigma$ denotes the forward jump, and $\hat{f}$ denotes the solution of the shifting problem.