I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below.
My attempt: For simplicity, assume everything is one-dimensional, with one risky asset $S$ and a risk-less asset $S^0$. Assume also that $r$, $\sigma$ and $\mu$ are constants. For brevity, I have skipped some of the computations (hopefully they are correct). Assume a probability space with a real-world measure $P$. The dynamics of the risky asset price is $d S_t = \mu S_t dt + \sigma S_t dW_t^P$ for $W^P$ a $P$-Brownian motion. Define the discounted asset price $\tilde{S}_t = S_t/S_t^0 = e^{-rt} S_t$.
For the first implication, assume that $d S_t = S_t r dt + S_t \sigma d W_t^Q$. Itô's formula then gives that $d \tilde{S}_t = \tilde{S}_t \sigma d W_t^Q$, so $\tilde{S}$ is an Itô integral hence a martingale, thus $Q$ is a martingale measure.
For the second implication, assume that $Q$ is a martingale measure. By Itô's formula again, $d \tilde{S}_t = \tilde{S}_t((\mu-r)dt + \sigma dW_t^P) \ (\ast)$. Since $Q$ is a martingale measure, by Girsanov's theorem $dW_t^P = dW_t^Q - \frac{\mu-r}{\sigma}dt$ for $W^Q$ a $Q$-Brownian motion, which inserted into $(\ast)$ yields $d \tilde{S}_t = \tilde{S}_t \sigma W_t^Q$. ''Reversing'' Itô's formula with respect to this last expression, yields $d S_t = S_t r dt + S_t \sigma dW_t^Q$, hence $r = \mu$.
Is this correct? My problem is that Girsanov's theorem is not covered prior to this result in the book (I think). Should the result follow without it?