I am confused about the solution given here to part c of the question posed below. I understand the algebra steps, but I do not understand how we can conclude at the end that therefore A_t and R_t are independent. I thought to show independence of the CDFs, we would need to show that $P(A_t \leq x, R_t \leq y) = P(A_t \leq x)\cdot P(R_t\leq y)$. I do not see how the result derived, that $P(A_t > x, R_t > y) = P(A_t > x)\cdot P(R_t > y)$, implies independence. I believe I am missing some useful identity in probability.
And the solution for part c is given as follows:
\begin{align} &P\big(A_t\le x, R_t\le y\big)\\ &=P\big(R_t\le y\big)-P\big(A_t>x, R_t\le y\big)\\ &=P\big(R_t\le y\big)-\big(P\big(A_t>x)-P\big(A_t>x,R_t>y\big)\big)\\ &=P\big(R_t\le y\big)-1+P\big(A_t\le x\big)+P\big(A_t> x\big)P\big(R_t> y\big)\\ &=P\big(R_t\le y\big)-1+P\big(A_t\le x\big)\\ &\hspace{2em}+\big(1-P\big(A_t\le x\big)\big)\big(1-P\big(R_t\le y\big)\big)\\ &=P\big(A_t\le x\big)P\big(R_t\le y\big) \end{align}