Define the quadratic variation of a semimartingale $(X_t)_{t \geq 0}$ by
$$[X,X]_t := \mathbb{P}-\lim_{n \to \infty} \sum_{j=0}^n (X_{t_j}-X_{t_{j-1}})^2$$
where $\Pi_n := \{t_0<\ldots<t_n<t\}$ is a sequence of partitions such that $|\Pi_n| \to 0$. Moreover, we set $$[X,Y]_t := \frac{1}{4} ([X+Y]_t-[X-Y]_t).$$ Where $Y_t$ is another semimartingale. How does one prove from this definition that
$$[X,Y]_t :=\mathbb{P}- \lim_{n \to \infty} \sum_{j=0}^n (X_{t_j}-X_{t_{j-1}})(Y_{t_j}-Y_{t_{j-1}})$$?
Write down the definition for $[X+Y]_t$ and $[X-Y]_t$ and then consider that
$$\left((X + Y)_{t_j}-(X+Y)_{t_{j-1}}\right)^2 - \left((X-Y)_{t_j}-(X-Y)_{t_{j-1}}\right)^2 = 4(X_{t_j}-X_{t_{j-1}})(Y_{t_j}-Y_{t_{j-1}})$$ by using $$(X + Y)_{t_j} = X_{t_j} + Y_{t_j} \\ (X - Y)_{t_j} = X_{t_j} - Y_{t_j}$$