Corollary 2

Let $X = (X_i)_{i \in I}$ and $Y = (Y_i)_{i \in I}$ be two families of indeterminates. The set of polynomials $u \in A[X,Y]$ such that $u(X,X) = 0$ is the ideal of $A[X,Y]$ generated by the polynomials $X_i - Y_i$ (for $i \in I$).

This corollary results directly from Cor. 1 on replacing $A$ by $A[Y]$ and $a_i$ by $Y_i$, interpreting $A[X,Y]$ as polynomial ring in the $X_i$ with coefficients in $A[Y]$.