Smooth Local Coordinates for Product Manifold

Theorem
Let $M_1, M_2$ be Riemannian manifolds.

Let $\tuple {x^1, \dots, x^n}$ be the smooth local coordinates for $M_1$.

Let $\tuple {x^{n + 1}, \dots, x^{n + m}}$ be the smooth local coordinates for $M_1$.

Let $M_1 \times M_2$ be the product manifold.

Then the smooth local coordinates for $M_1 \times M_2$ can be written as $\tuple {x^1, \dots, x^{n + m}}$.