The regularity index, Hilbert function of fat points and the embedding

Let $m \ge n$, $\phi_{n,m}: \mathbb P^n \to \mathbb P^m$, $\phi_{n,m}(a_1, \ldots, a_n)=(a_1, \ldots, a_n, 0, \ldots, 0)$, be the embedding, $Z=m_1P_1+\cdots+m_sP_s$ be fat points in $\mathbb P^n$ and $\phi_{n,m}(Z)=m_1\phi_{n,m}(P_1)+\cdots+m_s\phi_{n,m}(P_s)$ be fat points in $\mathbb P^m$. We show the relation between the regularity index, Hilbert function of the fat points $Z$ and the regularity index, Hilbert function of the fat points $\phi_{n,m}(Z)$.