Prove that product charts overlap smoothly

Question

If M and N are manifolds, let
$$\varepsilon=(x^1,...,x^m):\mathscr U\rightarrow R^m \ and \ \gamma=(y^1,...,y^n):\mathscr V\rightarrow R^n$$ be coordinate systems (charts) in M and N, respectively. The product function
$\varepsilon \times \gamma:\mathscr U \times \mathscr V\mathscr \ \rightarrow \ R^{m+n} \ $ is defined by
$$(\varepsilon \times \gamma)(p,q) = (x^1(p),...,x^m(p),y^1(q),...,y^n(q)).$$ Evidently $\ \varepsilon \times \gamma \ $ is a coordinate system in the Hausdorff space $\mathit M \times \mathit N$.
But how can I prove that product charts (similar to $ \ \varepsilon \times \gamma \ $) on $\mathit M \times \mathit N$ overlap smoothly.
Thank you in advance.

Answer 1

Write out what the transition functions will be on the product and use the fact that the the original transition functions are smooth.