Prove this statement is the cash value of a death insurance policy on the second life.
$$ A_{\overline{xy}} = A_x + A_y - A_{xy} = 1 - d \cdot \ddot{a}_{\overline{xy}} $$
I can't find the proof anywhere, the statement seems logical but I don't know how should I go about proving it, can someone help me? Show that the present value of a death insurance for the second life is calculated according to the equation with the connecting pension to the last life.
$$ \ddot{a}_{\overline{xy}} = \sum_{k=0}^\infty v^k \cdot {}_kp_{\overline{xy}} $$
$$ A_{\overline{xy}} = \sum_{k=0}^\infty v^{k+1} \cdot {}_{k|}q_{\overline{xy}} = \sum_{k=0}^\infty v^{k+1} \cdot \left( {}_{k|}q_x + {}_{k|}q_y - {}_{k|}q_{xy} \right) $$
The first equality is obvious. $$A_{xy} + A_{\overline{xy}}$$ represents the actuarial present value of a whole life insurance of $1$, payable at the end of the policy year of the first death of the joint status $(xy)$, plus the same, payable at the end of the year of the second death of the joint status $(xy)$. Since there are only two insureds, this is equivalent to $$A_x + A_y$$ where the statuses are explicitly identified. In other words, if an insurer offers a policy that pays upon the death of $(x)$, and again pays upon the death of $(y)$, then this is the same as the insurer offering a policy where they pay upon the first death, and then again on the second death.
The analogous mathematical property is $$\{x, y\} = \{\min(x,y), \max(x,y)\}.$$
The formula you wrote also makes this relationship evident:
$$A_{\overline{xy}} = \sum_{k=0}^\infty v^{k+1} \cdot {}_{k|}q_{\overline{xy}} = \sum_{k=0}^\infty v^{k+1} \cdot \left( {}_{k|}q_x + {}_{k|}q_y - {}_{k|}q_{xy} \right) $$ and all you have to do is distribute $v^{k+1}$ and write each sum separately: for instance, $$A_x = \sum_{k=0}^\infty v^{k+1} {}_{k|}q_x,$$ and so forth.
As for the second equality, I will leave it as an exercise, since you have not shown your own effort. Why don't you try to relate the first formula to the second? What do you know about the relationship between a life annuity and life insurance?