Unable to understand this expression

Question

I am unable to understand the above expression from the 3rd line onwards. Why is P(A|q1...qt-1)P(q1..q-1) = P(A|qt-1)P(q1..q-1)

I am not clear of the subsequent line as well

Answer 1

This follows from the Markov property (or the memoryless property) that the conditional probability of a future event only depends on the current state.

See link for more information about this property.