Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly regular pair. In this paper, we provide the same improvement for a jointly regular family: let h : ${mathbb{P}}_{mathbb{Q}}^n{ ightarrow}{{mathbb{R}}$ be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is de ned in [10] and let {$f_1,{ldots},f_k|f_l:mathbb{A}^n{ ightarrow}mathbb{A}^n$} bbe finite set of polynomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of $f_1,{ldots},f_k$ is empty, then there is a constant C such that $ sumlimits_{l=1}^kfrac{1}{def;f_iota}h(f_iota(P))>(1+frac{1}{r})f(P)-C$ for all $P{in}mathbb{A}^n$ where r= $max_{iota=1},{ldots},k(r(f_l))$.