Proof that if composition of a curve $\gamma$ and a regular surface is a regular curve, than the curve $\gamma$ is regular

Question

I have a curve $\gamma : A \subset \mathbb{R} \to D \subset \mathbb{R}^2$ and a regular surface $ S:D \subset \mathbb{R}^2 \to \mathbb{R}^3$. Their composition $S \circ \gamma$ is a regular curve in $\mathbb{R}^3$. Can I state that $\gamma$ is a regular curve?

Answer 1

Yes, because $\gamma$ is the composition of a diffeomorphism $S(D) \rightarrow D$ with a regular curve.