$\displaystyle (A*B)^t = B^t * A^t$

So, my problem is, how to proof this is true?

What I could do:

Lets Say $\displaystyle A axb$ and $\displaystyle B bxc$, so, in this case, $\displaystyle AB axc$, and obviously $\displaystyle AB^T cxa$. The only way of getting $\displaystyle AB^T cxa$ is multiplying $\displaystyle B^t cxb$ for $\displaystyle A^t bxa$, which would be false if we used $\displaystyle A^t bxa * B^t cxa$, since $\displaystyle a =/= c$. Even though this makes sense, is this a valid proof? And is there a cleaner one?

PS: A and B are matrices.