Products of Embedded Submanifolds

[Arkuski]

I'm trying to come up with a simple proof that if $M$ is an embedded submanifold of $N$, and $P$ is an embedded submanifold of $Q$, then $M×P$ is an embedded submanifold of $N×Q$. I'm thinking this could be easily done using showing that $M×P$ satisfies the local $k$-slice condition, or that the product of smooth embeddings (from the respective inclusion maps) is also a smooth embedding.

 

[R136a1]

Yes, both the slice-condition as the map-condition work. But what did you try? Where are you stuck?