A remark on $\mathscr C^\infty $ definable equivalence

We establish that if a submanifold $M$ of $\mathbb R^n$ is definable in some o-minimal structure then any definable submanifold $N\subset \mathbb R^n$ which is $\mathscr C^\infty $ diffeomorphic to $M$, with a diffeomorphism $h:N\to M$ that is sufficiently close to the identity, must be $\mathscr C^\infty $ definably diffeomorphic to $M$. The definable diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood of $M$.