The theorem for discrete-time martingales is as follows:
Let $X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$ be a supermartingale and $\tau_1,\tau_2$ two a.s. bounded stopping times on the filtration $(\mathcal{F}_n)_n$ with $\tau_1\leq \tau_2$ a.s. Then the r.v.'s $X_{\tau_1}$ and $X_{\tau_2}$ are integrable and $E(X_{\tau_2}\ |\ \mathcal{F}_{\tau_1} )\leq X_{\tau_1}$
Probably there's something I'm not aware of, because it seems too much obvious to me: By definition of martingale $X_n$ has to be integrable for every $n$, so it should be for $\tau_n$ too. And also by definition of supermartingale shouldn't $E(X_{\tau_2}\ |\ \mathcal{F}_{\tau_1} )\leq X_{\tau_1}$ be true just because $\tau_1\leq \tau_2$?