Birational map between product of projective varieties What is an example of a birational morphism between $\mathbb{P}^{n} \times \mathbb{P}^{m} \rightarrow \mathbb{P}^{n+m}$?

The subset $\mathbb A^n\times \mathbb A^m$ is open dense  in $\mathbb P^n\times \mathbb P^m$ and the subset  $\mathbb A^{n+m}$ is open dense in $\mathbb P^n\times \mathbb P^m$.
Hence the isomorphism $\mathbb A^n\times \mathbb A^m\stackrel {\cong}{\to} \mathbb A^{n+m}$ is the required birational isomorphism.  
The astonishing point is that a rational map need only be defined on a dense open subset , which explains the uneasy feeling one may have toward the preceding argument, which may look like  cheating.
The consideration of "maps" which are not defined everywhere is typical  of algebraic ( or complex analytic) geometry, as opposed to other geometric theories like  topology, differential geometry,...
