I am reading about formal group homomorphisms defined over a ring $R$ of characteristic $p > 0$ from Silverman's Arithmetic of Elliptic Curves. Given a homomorphism $f$, he shows that if $f'(0)=0$ then $\text{ht}(f)\geq1$ and if $f(T) = g(T^{p^h})$ with $h = \text{ht}(f)$, then $g'(0) \neq 0$.
Using these lemmas, he attempts two prove that given formal group law homomorphisms, $f:\mathcal{F}\rightarrow\mathcal{G}$ and $g:\mathcal{G}\rightarrow\mathcal{H}$, $$\text{ht}(f\circ g) = \text{ht}(f) + \text{ht}(g)$$
Writing $f$ and $g$ as $f(T) = f_1(T^{\text{ht}(f)})$ and $g(T) = g_1(T^{\text{ht}(g)})$ respectively, with both $g'(0),f'(0)\neq 0$ by the above lemma. Now, $(f\circ g)(T) = g_1(f_2(T^{\text{ht}(f)+\text{ht}(g)}))$, where $f_2$ is just $f_1$ with each coefficient raised to $p^{\text{ht}(g)}$. But then he concludes that this is enough to see the result. I see that this gives me that $\text{ht}(f\circ g) \geq \text{ht}(f) + \text{ht}(g)$, but I don't see how to obtain the equality. If we additionally assume that $R$ is an integral domain, then using $f'(0),g'(0)\neq 0$, we can obtain that $(f\circ g)'(0)\neq 0$ which implies the result. But Silverman does not assume that, and without it, I don't really see how to use $g'(0),f'(0)\neq 0$.
This is Proposition IV.7.3 from Silverman's Arithmetic of Elliptic Curves, page 133 in the second edition.