Problem 1-A

Let $M_1\subset R^A$ and $M_2\subset R^B$ be smooth manifolds. Show that $M_1\times M_2\subset R^A\times R^B$ is a smooth manifold, and that the tangent manifold $D(M_1\times M_2)$ is canonically diffeomorphic to the product $D{M}_1\times D{M}_2$. Note that a function $x\mapsto (f_1(x),f_2(x))$ from $M$ to $M_1\times M_2$ is smooth if and only if both $f_1:M\to M_1$ and $f_2:M\to M_2$ are smooth.