Algebraically Independence and Algebraic Extension

Question

If $A=K[X_1,\dots,X_n]$ is the polynomial ring over $K$, and $\{f_1,\dots, f_n\}\subset A$ such that $K(X_1,\dots, X_n)$ is algebraic over $K(f_1,\dots,f_n)$, then is $\{f_1,\dots f_n\}$ algebraically independent over $K$?

It is better if this can be done without some other lemmas or theorems.