Let $\mathbf{P}^{n}$, $\mathbf{P}^{m}$ be two projective spaces over an algebrically closed field of characteristic p; let $\varphi : \mathbf{P}^{n} \to \mathbf{P}^{m}$ be a rational map; let $\mathcal{V}$ be a subvariety of $\mathbf{P}^{n}$. 1) If $p = 0$, there exists a biregular map $\Phi : \mathbf{P}^{n} \to \mathbf{P}^{N}$, with $\mathbf{P}^{N} \supset \mathbf{P}^{m}$ and $N \le m+n+1-\operatorname{dim} \operatorname{Im}\varphi$, such that $\Phi \mid_\mathcal{V} = \varphi \mid_{\mathcal{V}}$. 2) If $\varphi \mid_{\mathcal{V}}$ is biregular, there exists, whatever p may be, $\Phi' : \mathbf{P}^{n} \to \mathbf{P}^{N}$ biregular, with $N \le m+n+1 - \operatorname{dim} \mathcal{V}$, such that $\Phi' \mid_{\mathcal{V}} = \varphi \mid_{\mathcal{V}}$.