There is a theorem which says that number of sequences $(x_1, \ldots, x_k, \ldots, x_n)$ where $x_k$ can be choosen in $m_k$ ways, $k = 1, 2, \ldots, n$ is equal to $m_1 \cdot m_2 \cdot \ldots \cdot m_n$.
I'm looking for a proof (or proofs) and official name of this theorem.
Thank you for your help.