I am reading a proof in which I don't understand how to use Ito's rule to derive the following:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are continuous local martingales null at $0$. Let $b_i, b_k : [0,\infty) \times \mathbb{R}^d \to \mathbb{R}$ be Borel-measurable maps and let $X$ be a process taking values in $\mathbb{R}^d$. Then it claims that \begin{eqnarray} && \int_0^t (M^{(i)}_s - M^{(i)}_t) b_k (s, X_s) \,ds + \int_0^t (M^{(k)}_s - M^{(k)}_t) b_i (s, X_s) \,ds \\ &= & - \int_0^t \bigg[ \int_0^s b_k (u,X_u) \,du \bigg] \, dM^{(i)}_s -\int_0^t \bigg[ \int_0^s b_i (u,X_u) \,du \bigg] \, dM^{(k)}_s. \end{eqnarray}
Any idea of how to show this? The book claims that it requires an application of Ito's formula but I do not know what expression should I apply it to.