Delta partial derivative of shift along diagonal

Theorem

If the solution of the shifting problem $\hat{f}$ has partial $\Delta$-derivatives of all orders, then $$\dfrac{\partial^k \hat{f}}{\Delta^k t} (t,t)=f^{\Delta^k}(t_0),$$ where $\dfrac{\partial^k}{\Delta^k t}$ denotes a partial delta derivative.