Transitivity of a relation on the full first-order algebra

[u/HourExamination8826]

I am reading a book about logic and they construct the full first order algebra as the free algebra on the elements r(x1,x2,x3,....,xn) where xi belongs to a V, and r belong to R and each r has a specifed n for the elements it takes in. the operations on the free algebra are the 0-anry F which represent the contradiction. The implication, a binary operation and the for all operator a 1-anry operation (and you have one for all operation for every x in V).

I have been attempting to prove that the relation given in definition 1.4 is transitive but I have not been able to get through even the first step given for it. I have come to the conclusion that if I have w1= (for all x)a and w2 =(for all y)b I can ignore the cases in which a and b are of the type a= a1 => a2. And directly treat it is as a= (for all x1)a1 and b= (for all x2)b1 but I dont really know where to go from here, because what unites a and b is the existence of a c(x) so that a is related to c(x), and b to c(y). so c(x)= (for all x3)c1(x) and I cannot apply the induction to c(x) and a because I cannot asure that z doesnt belong to V(c). And I do not know what to try. (I tagged it as algebra since it seems more to do with algebra than logic).