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Well, this follows from the fact, that $Z$ inherits the martingale (w.r.t $Q$) property from $Z^1$ and $Z^2$. Indeed, for fixed $t$ we have to prove for $s\le u$
$$E[Z_u|\mathcal{F}_s]=Z_s$$
Where the expectation is taken with respect to $Q$. There are three different cases:
$\mathbf{Case: t\le s\le u}$
$$E[Z_u|\mathcal{F}_s]=E[\mathbf1_AZ^1_u|
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