optimal stopping finite horizon dynamic programming principle

Question

Suppose we are in the setting of optimal stopping in the finite horizon N discrete finite time framework with gain functions $G_{1 \leq n \leq N}$ measurable with respect to an underlying filtration $F_{n}$ and we are interested in maximizing $ \mathbb E\left(G_{\tau}\right)$ over all stopping times $ 1 \leq \tau \leq N$ . Let denote: $ V\left(n\right):=\sup\mathbb E\left( G_{\tau_{n}}\right)$, where the sup is taken over all stopping times $ n \leq \tau_{n} \leq N$. I wonder if the following holds true: $$ V(n)=\underset{\sigma}\sup\mathbb E\left(\mathbb 1_{ \sigma<\tau}G_{\sigma}+\mathbb 1_{\sigma \geq \tau}V(\tau)\right)$$, where the sup is taken over all stopping times $ n \leq \sigma \leq N$ ; and $n \leq \tau \leq N$ being any arbitrary stopping time.