A universal coefficient theorem for gauss's lemma

We shall prove a version of Gauß's lemma. It works in Z[a,A, b,B] where a = {a_i}^m _i=0, A = {A_i}^m _i=0, b = {b_i}ⁿ _j=0, B = {B_j}ⁿ _j=0, and constructs polynomials {c_k}k=0,... ,m+n of degree at most in each variable set a,A, b,B, with this property: setting for elements a_i,A_j, b_j, B_j in any commutative ring R satisfying, the elements ck = c_k(a_i,A_i, b_j,B_j) satisfy.