= Solution
If $M,N$ are <weakly orthogonal continuous martingales>, taking the two times equal gives $\mathbb E[M_sN_s]=0$. Conversely suppose this equal-time condition holds. For $s\le t$, the <martingale> property and <conditional expectation> give
$$
\mathbb E[M_sN_t]=\mathbb E[M_s\mathbb E(N_t\mid\mathcal F_s)]
=\mathbb E[M_sN_s]=0.
$$
For $t\le s$, condition the other factor at time $t$ instead. Products are integrable by the <Cauchy-Schwarz inequality>. Thus the equal-time condition implies the full cross-time definition, proving (i) equivalent to (ii).
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