Product Manifolds and Tangent spaces

Question

Let $M\subset{E^{n}}$ be an r manifold and $N\subset{E^{m}}$ be an s manifold. Regarding $E^{m+n}$ as the Cartesian product $E^{n}\times{E^{m}}$, show that $M\times{N}$ is an (r+s)manifold. Show that the tangent space at a point of $M\times{N}$ is the Cartesian product of the tangent spaces at the corresponding points of M and N. Can any one help me to do this problem.