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\label{thm:composition-closure}
$\mathrm{HolFun}$ is closed under composition.
If $S, T \in \mathrm{HolFun}$,
then $S \circ T \in \mathrm{HolFun}$.

We must verify both conditions
of the HolFun definition (I.D47).

\emph{Condition (1): D-holomorphy.}
$S \circ T$ is D-holomorphic
because the composition of D-holomorphic functions
is D-holomorphic.
Explicitly:
if $S(u, v) = (s_1(u),\; s_2(v))$
and $T(u, v) = (t_1(u),\; t_2(v))$
in sector coordinates (by I.P22), then
\[
    (S \circ T)(u, v)
    = S(t_1(u),\; t_2(v))
    = (s_1(t_1(u)),\; s_2(t_2(v))),
\]
which is again sector-independent:
the $B$-sector output $s_1(t_1(u))$
depends only on the $B$-sector input $u$,
and the $C$-sector output $s_2(t_2(v))$
depends only on the $C$-sector input $v$.
The split-CR equations are preserved under composition
because the chain rule holds in sector coordinates.

\emph{Condition (2): Tower coherence.}
For all $k \leq \ell$,
we must show that
$\pi_{\ell \to k}((S \circ T)(x))
= (S \circ T)(\pi_{\ell \to k}(x))$.
We compute:
\begin{align*}
    \pi_{\ell \to k}((S \circ T)(x))
    &= \pi_{\ell \to k}(S(T(x))) \\
    &= S(\pi_{\ell \to k}(T(x)))
    && \text{(by $S$'s tower coherence)} \\
    &= S(T(\pi_{\ell \to k}(x)))
    && \text{(by $T$'s tower coherence)} \\
    &= (S \circ T)(\pi_{\ell \to k}(x)).
\end{align*}
Both conditions are satisfied,
so $S \circ T \in \mathrm{HolFun}$.