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?