Yes, both equations hold.
A relatively simple proof is to note that for any random variable \(R\),
$$\ddot{a}_\overline{R|}^{m} = \frac{1-v^{R}}{d^{(m)}} $$
Letting \(R = T_x\), implying that \(m = \infty \), or \(R = K_x^{(m)} + \frac{1}{m} \), and taking the expectation on both the left and right and side of each of the equations yields the desired results respectively.
A more detailed proof can be found on pages 116 and 120 of DHW, \(2^{nd}\) edition.