p₁ = 0,…,p_r = 0, d1 ≠ 0,…,d₃ ≠ 0

where p_i and d_i are differential polynomials in K{u₁,…, u_m, x₁ ,…, x_n} and the u are parameters. According to this theorem we can identify all parametric values for which the parametric system has solutions for the x_i and at the same time giving the solutions for the x_i in an explicit way, i.e., the solutions are given by differential polynomial sets in triangular form. In the algebraic case, i.e. when p_i and d_i are polynomials, we present a refined algorithm with higher efficiency. As an application of the zero structure theorem presented in this paper, we give a new algorithm of quantifier elimination over differential algebraic closed fields. The algorithm has been implemented and several examples reported in this paper show that the algorithm is of practical value.