Rational Variant of Hartogs' Theorem on Separate Holomorphicity

Given an uncountable algebraically closed field $K$, we proved that if partially defined function $f\colon K \times \dots \times K \dashrightarrow K$ defined on a Zariski open subset of the $n$-fold Cartesian product $K \times \dots \times K$ is rational in each coordinate whilst other coordinates are held constant, then $f$ is itself a rational function in $n$-variables.