Let $(S_n)$ a martingale refer to $(X_n)$. Show that for all integer $k\leq l\leq m$ $$\mathbb E[(S_m-S_l)S_k]=0.$$
I don't understand the to following equality: $$\mathbb E[(S_m-S_l)S_k]=\mathbb E\big[\mathbb E[(S_m-S_l)S_k\mid X_k,...,X_1]\big]=\mathbb E\big[S_k\mathbb E[(S_m-S_l)\mid X_k,...,X_1]\big].$$
Which properties is used ?
The first equality is always true. The second is the definition of the conditional expectation, plus the fact that $S_k$ is a deterministic function of $$ X_k \dots X_1 $$