I was reading a book about Equations for the Expected Time to Absorption, it says:
The expected times to absorption, $a_1,....a_M$ are unique solutions to the equations $a_i=0$ for all absorbing states $i$ and $$a_i = 1+\sum_{j=1}P_{ij}a_{j}$$
Note: $P_{ij}$ represents the transition probability from i to j
For all transient states $i$ These equations can be easily derived using the law of total expectation by conditioning the expected times to absorption on the next state.
So my question is: how can we prove the above statement true using law of total expectation?