[proofplan]
We reduce the additive Cousin problem to the vanishing of the first cohomology group of the structure sheaf. The structure sheaf $\mathcal{O}_X$ is coherent by Oka's Coherence Theorem. Since $X$ is Stein, Cartan's Theorem B annihilates the higher cohomology of every coherent analytic sheaf on $X$, in particular $H^1(X,\mathcal{O}_X)$. Finally, the Čech interpretation of $H^1(X,\mathcal{O}_X)$ identifies this vanishing with solvability of every additive Cousin datum.
[/proofplan]
[step:Apply Oka coherence to the structure sheaf]
Let $\mathcal{O}_X$ denote the sheaf of holomorphic functions on the complex manifold $X$. By Oka's Coherence Theorem, the sheaf $\mathcal{O}_X$ is a coherent analytic sheaf on $X$.
[guided]
We first identify the sheaf to which Cartan's Theorem B will be applied. Let $\mathcal{O}_X$ denote the sheaf assigning to each open set $U \subseteq X$ the ring $\mathcal{O}_X(U)$ of holomorphic functions $U \to \mathbb{C}$. The theorem needed here is Oka's Coherence Theorem, whose conclusion is precisely that the structure sheaf $\mathcal{O}_X$ of a complex manifold is coherent as an analytic sheaf. Thus $\mathcal{O}_X$ satisfies the coherence hypothesis required by Cartan's Theorem B.
[/guided]
[/step]
[step:Use Cartan's Theorem B on the Stein manifold $X$]
Since $X$ is Stein by hypothesis and $\mathcal{O}_X$ is coherent, Cartan's Theorem B applies to $\mathcal{O}_X$. Therefore, for every integer $q \geq 1$,
\begin{align*}
H^q(X,\mathcal{O}_X) = 0.
\end{align*}
Taking $q = 1$ gives
\begin{align*}
H^1(X,\mathcal{O}_X) = 0.
\end{align*}
[guided]
Cartan's Theorem B states that if $X$ is a Stein manifold and $\mathcal{F}$ is a coherent analytic sheaf on $X$, then
\begin{align*}
H^q(X,\mathcal{F}) = 0
\end{align*}
for every integer $q \geq 1$. We verify the two hypotheses. First, $X$ is Stein by the theorem statement. Second, the sheaf $\mathcal{O}_X$ is coherent by the previous step, using Oka's Coherence Theorem. Hence Cartan's Theorem B applies with $\mathcal{F} := \mathcal{O}_X$, and it yields
\begin{align*}
H^q(X,\mathcal{O}_X) = 0
\end{align*}
for every integer $q \geq 1$. In particular, choosing $q = 1$ gives
\begin{align*}
H^1(X,\mathcal{O}_X) = 0.
\end{align*}
[/guided]
[/step]
[step:Translate the cohomology vanishing into solvability of additive Cousin data]
Let $\{U_i\}_{i \in I}$ be an open cover of $X$, and let
\begin{align*}
g_{ij}: U_i \cap U_j \to \mathbb{C}
\end{align*}
be holomorphic maps satisfying the additive Čech cocycle condition
\begin{align*}
g_{ij} + g_{jk} + g_{ki} = 0
\end{align*}
on every triple intersection $U_i \cap U_j \cap U_k$. This collection defines a class $[g] \in H^1(X,\mathcal{O}_X)$. Since $H^1(X,\mathcal{O}_X)=0$, the class $[g]$ is zero, so there exist holomorphic maps
\begin{align*}
h_i: U_i \to \mathbb{C}
\end{align*}
such that
\begin{align*}
g_{ij} = h_j - h_i
\end{align*}
on $U_i \cap U_j$. This is exactly the solvability condition for the additive Cousin problem. Hence every additive Cousin problem on $X$ is solvable, and equivalently
\begin{align*}
H^1(X,\mathcal{O}) = 0.
\end{align*}
[/step]
