Expected Value of a function of a discrete Random Variable. Help understanding the proof?

Question

I don't understand why the double summation shown next to the red star) in the picture of my lecturer's notes collapses to the one summation in the following line. Can someone please explain it to me?

Answer 1

The answer lies in the item that follows: "(i) each $x_k$ appears exactly once in both expressions".

Maybe doing it the other way round would help. You have a sum of the form $\sum_k h(x_k)$. For all $n$, let $A_n=\{k\mid g(x_k)=y_n\}$ be the set of elements $k$ such that $g(x_k)=y_n$. Since each $k$ belongs to one (and only one) $A_n$, we have $$ \sum_k h(x_k)=\sum_n\sum_{k\in A_n}h(x_k)=\sum_n\sum_{k:g(x_k)=y_n}h(x_k). $$