= Solution
Fix $0\leq r\leq t$. Since $M$ is a <martingale>,
$$
\operatorname{Cov}(M_{t+s}-M_t,M_r)=\mathbb E\!\left[M_r\,\mathbb E[M_{t+s}-M_t\mid\mathcal F_t]\right]=0.
$$
Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\ldots,M_{r_n}$ has a <multivariate normal distribution>. Therefore <uncorrelated jointly normal variables are independent>, so the increment is independent of every finite vector of past values. A <Monotone class theorem> then extends this to independence from $\mathcal F_t=\sigma(M_r:0\leq r\leq t)$. This is the <independent increments of a Gaussian martingale>.
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