I want to define a Riemann metric using the functions $g_{ij}$ on local coordinate system $(x^1,x^2,...,x^n)$ on $U$ and functions $h_{ij}$ on local coordinate system $(y^1,y^2,...,y^n)$ on $U$ and I have managed to understand the relationships between the functions $g_{ij}$ and $h_{ij}$.
How can I prove that being the functions $g_{ij}$ from the class $C^r$ on $U$ is independent on choosing the local coordinate system ???