In the beginning of page 120, to establish the formal group law for an elliptic curve, the book adds 2 points $(z_1,w_1)$ and $(z_2,w_2)$, where $w_1 = w(z_1), w_2 = w(z_2)$ using the group law. It makes explicit calculation, and argues that the the coordinates $z_3$ and $w_3$ of the inverse of the sum will be in $Z[a_1,a_2,...a_6][[z_1,z_2]]$.
Then, it uses Proposition 1.1 chapter IV, page 116 saying that the formal power series $w(z)$ is the unique power series satisfying $w(z) = f(z,w(z))$, hence $w_3 = w(z_3)$. However, the proposition requires that the coefficients of the power series $w(z)$ are in $Z[a_1,a_2,...a_6]$, instead of $Z[a_1,a_2,...a_6][z_1,z_2]$, which is what we need. Can anyone help me resolve this problem please?
The fact that $w_3=f(z_3,w_3)$ implies (as in the argument in p. 116) that $w_3$ is a power series in $\mathbb{Z}[a_1,a_2,\ldots,a_6][[z_3]]$. Treating $z_3$ as an indeterminate, together with the fact that $w(z)$ is the unique power series with $w(z)=f(z,w(z))$ allows us to conclude that $w_3=w(z_3)$ as claimed.