Corollary 1

Let U be an open neighbourhood of K. Let $u_1$ and $u_2$ be continuous mappings of $\mathscr{O}(U; A)$ into a topological space X factoring through $\mathscr{O}(K; A)$. Then $u_1=u_2$ if and only if $u_1$ and $u_2$ coincide on the set of restrictions to K of polynomial functions on $\mathbf{C}^n$ with coefficients in A.