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Use just the parameters $1$ and $-1$. Positivity and adaptation of their local martingales giveThus both processes are adapted. A positive continuous local martingale has a semimartingale logarithm, by Itô formula after localization away from zero. Since $X=\log Z^{(1)}+A/2$ and $A$ has finite variation, $X$ is a continuous semimartingale. Write $X=N+V$, where $N$ is a zero-starting continuous local martingale and $V$ is continuous finite variation, also starting from zero.

Apply Itô formula to each exponential:Because $Z^{(\theta )}$ is itself a local martingale, the finite-variation term is zero by part (a). Divide its finite signed measure by the strictly positive $Z^{(\theta )}$. For $\theta =1,-1$, respectively,Adding and subtracting show $V=0$ and $[N]=A$. Therefore the exponential criterion for a continuous local martingale and its bracket givesThe semimartingale property was established before applying Itô's formula to $X$; it was not assumed from the outset.