Theorem I.7.5 (Hilbert-Serre) Hartshorne uniqueness of Hilbert polynomial.

Question

I don't understand why the Hilbert polynomial is unique. In the theorem I.7.5 we find a polynomial $P_M(z)$ such that $\varphi_M(l)=P_M(l)$ for all $l\gg0$ ($\varphi_M(l)=dim_K M_l$ as discribed in the definition right before the theorem) and $deg P_M= dim V(Ann (M))$. It should be clear but I don't see why...