I'd like to show that
$$\langle I^M(X),I^N(Y)\rangle_t=\int_0^t X_uY_u\,d\langle M,N\rangle_u\text{ holds for all } t\ge0,\ P-a.s.$$ is equivalent to
$$E[(I_t^M(X)-I_s^M(X))(I_t^N(Y)-I_s^N(Y))|\mathcal F_s]=E\biggr[\int_s^t X_uY_u\,d\langle M,N\rangle_u|\mathcal F_s\biggr]\text{ holds for all }0\le s<t<\infty,\ P-a.s.$$
$M,N$ are continuous square integrable martingales, and $X,Y$ are progressively measurable processes for which the stochastic integrals $I^M(X)=\int_0^t X_s\,dM_s$ and $I^N(Y)=\int_0^t Y_s\,dN_s$ are defined (Definition 3.2.9 in Karatzas and Shreve's Brownian Motion and Stochastic Calculus, the text this question comes from).
It is easy to see that the former implies the latter, by noting that if $A,B$ are square integrable martingales and $0<t\le u<v$, $E[(A_v-A_u)(B_v-B_u)|\mathcal F_t]=E[\langle A,B\rangle_v-\langle A,B\rangle_u|\mathcal F_t].$ I'm having trouble seeing the other direction.
Any help is greatly appreciated. Thanks in advance.