# Time-ordered Derivative and Equal-time Commutator

In Green, Schwarz & Witten Superstring theory, Vol. I, page 141, I don't understand how pulling the derivative inside the Time-ordered product can give an Equal-time Commutator:

$$\tag{3.2.44} \partial_- \langle T \big( T_{++}(\sigma, \tau) T_{++}(\sigma', \tau') \big) \rangle ~=~ \frac12 \delta(\tau - \tau') \langle \big[ T_{++}(\sigma, \tau), T_{++}(\sigma', \tau) \big] \rangle$$

Is there any (rigorous) proof for this?

-

$$\mathrm{T}\left(A(t_1) B(t_2)\right) = \Theta(t_1 - t_2) A(t_1) B(t_2) + \Theta(t_2 - t_1) B(t_2) A(t_1).$$
$$\partial_{x} \Theta(x) = \delta(x).$$