Suppose $V$ is a finite-dimensional vector space and $R,S,T \in \mathcal L(V)$ are such that $RST$ is surjective. Prove that $S$ is injective.

By 3.69 $RST$ is invertible, apply 3d/9 twice to show $R,S$ and $T$ are all invertible, this shows $S$ is injective as a special case.

(Old direct proof)

By 3.69 $RST$ is invertible, clearly this requires $\text{null }T = \{0\}$ which shows $T$ is invertible by 3.69, similarly we must have $\text{range }R = V$ in order for $RST$ to be surjective meaning (again by 3.69) $R$ is invertible.

Suppose $Sv = 0$, then $RST(T^{-1}v) = 0$ implies $T^{-1}v = 0$ which implies $v = 0$ since $T$ is invertible. Thus $S$ is injective (and invertible by 3.69)

It wasn't in the question, but this shows that if $RST$ is invertible, then $R,S$ and $T$ are invertible and the inverse is given by $T^{-1}S^{-1}R^{-1}$.