In the book Manifolds, Tensors and Forms, there is the following theorem:
We have a short exact sequence $0\rightarrow \ker T\xrightarrow{\iota} V\xrightarrow{T} W\rightarrow 0$ where $\iota$ is the inclusion mapping and $T$ is surjective. If $S$ is a section of $T$, then $V=\ker T \oplus S(W)$.
In the proof, we want to show that $\ker T\cap S(W)=\{0\}$. To do so, he notes that $S$ is injective. Then he says that we have a unique $w\in W$ such that $S(w)=x$, and then using the fact that $S$ is a section we can conclude that $w=T\circ S(w)=T(x)=0$, and from the linearity of $S$ we must have $x=0$.
Now, this is all fine, but here's my question: why do we need the fact that $S$ is injective? Rephrased, why do we need that $w$ is unique? As far as I can tell, the argument doesn't depend on it, but only on the fact that $S$ is linear and that $S:W\to S(W)$ is surjective.