Recall that the increments of a Brownian motion are normally distributed with mean 0 and variance equal to the time increment. That is, $B(t) - B(s) \sim N(0, t-s)$.
Now, we can rewrite $B(s) + B(t)$ as $B(s) + (B(t) - B(s))$. Since $B(s)$ and $B(t) - B(s)$
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