Let $T$ be the time-ordering operator which orders operators $A_1(t_1), A_2(t_2), \ldots$ such that the time parameter decreases from left to right:
$$T[A_1(t_1) A_2(t_2)] = A_2(t_2) A_1(t_1) \text{ if } t_2 > t_1 \text{ and }= A_1(t_1)A_2(t_2) \text{ otherwise. } $$
The time $t_i$ does not have to be a physical time, it can also be an imaginary time, etc.
Question: I would like to know why the following equation holds: for $t_i \leq t_1, t_2 \leq t_f$ it holds that $$T\left[A_1(t_1) A_2(t_2) \exp\left(-i\int_{t_i}^{t_f}H(t) dt\right)\right] \\ = T\left[\exp\left(-i\int_{t_{\pi_1}}^{t_f}H(t) dt\right)\right] A_{\pi_1}(t_{\pi_1}) \cdot T\left[\exp\left(-i\int_{t_{\pi_2}}^{t_{\pi_1}}H(t) dt\right)\right] A_{\pi_2}(t_{\pi_2}) \cdot T\left[\exp\left(-i\int_{t_i}^{t_{\pi_2}}H(t) dt\right)\right] ,$$ where $\pi$ is a permutation such that the times are ordered.
I encountered this equation in Negele & Orland (1998) in eq. (2.49) on p. 63 and in eq. (2.67b) on p. 70, where they split the integral
$$\int_{t_i}^{t_f} dt = \int_{t_i}^{t_{\pi_2}} dt + \int_{t_{\pi_2}}^{t_{\pi_1}} dt + \int_{t_{\pi_1}}^{t_f} dt$$
and used the time-ordering. It appears in calculations of greens functions respectively correlation functions.
I tried to prove this equation in an elementary way by using
$$ T\left[\exp\left(-i\int_{t_i}^{t_f}H(t) dt\right)\right] = 1 + \sum_{n=1}^\infty \frac{(-i)^n}{n!} \int_{t_i}^{t_f} d\tau_1 \ldots \int_{t_i}^{t_f} d\tau_n T \left[ H(\tau_1) \ldots H(\tau_n) \right]$$
[cf. eq. (2.10) on p. 50] and applying the $T$-operator on the expression, but I did not succeed yet. If someone can show me a valid proof or point out some literature where it is proven, I'd be thankful.