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I know the formula below works

$$A_x=1-d\ddot{a}_x$$

I just want to ask if we do this for any type of whole life product. i.e. does the following hold:

$$\overline{A}_x=1-\delta \overline{a}_x$$

$$A^{(m)}_x=1-d^{(m)}\ddot{a}^{(m)}_x$$

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.