Take regular surfaces $M_1, M_2$ in $\mathbb{R}^3$ and suppose that we have the charts $x: U \rightarrow M_1 \qquad y: U \rightarrow M_2.$
Define the function $F=y\circ x^{-1}: x(U) \rightarrow y(U) $.
Why is $F$ a diffeomorphism between the surfaces $x(U)$ and $y(U)$?
Clearly $F$ is bijective, but I cannot see why $F$ is differentiable.