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