Prove Theorem 1.4.1.2, i.e., if \(X\) is continuous, \(X_{t}\) and \(X_{t}^{2}-t\) are martingales, then \(X\) is a BM.

Theorem 1.4.1.2:

The process \(X\) is an \(\mathbf{F}\)-Brownian motion if and only if the processes \(\left(X_{t}, t \geq 0\right)\) and \(\left(X_{t}^{2}-t, t \geq 0\right)\) are continuous \(\mathbf{F}\)-local martingales.