Good day to everyone!

My question concerns Hidden Markov Models and is pretty basic. In one of the books ("Introduction to Machine Learning" by Ethem Alpaydin, 2nd Edition, p.373), I get the following transition:

$P(O_1, \ldots, O_T \mid q_t = S) = P(O_1, \ldots, O_t \mid q_t = S) \cdot P(O_{t+1}, \ldots, O_T \mid q_t = S)$,

where $O_i$ are the public observations for HMM, and $q_t$ is the $t$-th hidden state, $S$ is some state. My question is why is the joint probability equals to product? Intuitively I understand that: we have conditioning on $q_t = S$, and with respect to it, the observations after $t$ are independent of the ones before $t$. But I can't prove that rigorously, having already wasted lots of paper :(

Thanks a lot for any hint!