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

Let $m\ge n$, ${\phi }_{n,m}:{\mathbb{P}}^n\to {\mathbb{P}}^m$, ${\phi }_{n,m}(a_1,\dots ,a_n)=(a_1,\dots ,a_n,0,\dots ,0)$, be the embedding, $Z=m_1{P}_1+\cdots +m_s{P}_s$ 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 ${\phi }_{n,m}(Z)$.