An Elimination Procedure

Let R be a differential ring and let $(\eta)$ be the following system of ordinary algebraic differential equations:

$(\eta)$={ P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0, R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in R\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $Q_{i}, R_{j} \in R\{ y \}$, ord(Q_i)=q_i and
ord(R_j)=r_j for all i and j.

First of all in the following subsubsection an elimination procedure of one differential variable will be shown.