Suppose that f₁, ¼, f_mf_1, \ldots , f_m satisfy functional equations of type¶¶ f_i(z^d) = P_i(z, f_i(z))     or     f_i(z) = P_i(z, f_i(z^d))f_i({z^d}) = P_i(z, f_i(z)) \quad {or} \quad f_i(z) = P_i(z, f_i({z^d})) ¶for i = 1, ¼, mi = 1, \ldots , m, an integer d > 1 and polynomials P_i Î \Bbb C (z)[ y]P_i \in \Bbb C (z)[ {y}] with pairwise distinct partial degrees deg_y( P₁), ¼, deg_y( P_m)\deg _y( {P_1}), \ldots , \deg _y( {P_m}). Generalizing a result of Keiji Nishioka and using an idea of Kumiko Nishioka we show, that f₁, ¼, f_mf_1, \ldots , f_m can only be algebraically dependent over \Bbb C (z)\Bbb C (z), if there is an index k Î { 1, ¼, m}\kappa \in \{ {1, \ldots , m}\} such that f_kf_{\kappa } is rational.