
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \(\mathrm{P}_{n}\) be a probability measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)\right)\), and let \(F_{n}=F_{n}\left(x_{1}, \ldots, x_{n}\right)\) be its corresponding distribution function, i.e.,

$$
F_{n}\left(x_{1}, \ldots, x_{n}\right)=\mathrm{P}_{n}\left(\left(-\infty, x_{1}\right] \times \ldots \times\left(-\infty, x_{n}\right]\right)
$$

Introduce the difference operator \(\Delta_{a_{i} b_{i}}: \mathbb{R}^{n} \rightarrow \mathbb{R}\left(a_{i} \leqslant b_{i}\right)\), acting according to the formula

$$
\Delta_{a_{i} b_{i}} F_{n}\left(x_{1}, \ldots, x_{n}\right)=F_{n}\left(\ldots, x_{i-1}, b_{i}, x_{i+1}, \ldots\right)-F_{n}\left(\ldots, x_{i-1}, a_{i}, x_{i+1}, \ldots\right)
$$

(all arguments except the \(i\)-th are fixed). Verify the validity of the equality

$$
\Delta_{a_{1} b_{1}} \ldots \Delta_{a_{n} b_{n}} F_{n}\left(x_{1}, \ldots, x_{n}\right)=\mathrm{P}_{n}\left(\left(a_{1}, b_{1}\right] \times \ldots \times\left(a_{n}, b_{n}\right]\right)
$$

Present the answer in LaTex format: \boxed{Your answer}
user