  Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$.
A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids
$M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an
$N$-minor. This is the third and final paper in a series where we prove that if
$|E(M)|-|E(N)| \ge 10$, then either $M$ has an $N$-detachable pair after
possibly performing a single $\Delta$-$Y$ or $Y$-$\Delta$ exchange, or $M$ is
essentially $N$ with a spike attached. Moreover, we describe the additional
structures that arise if we require only that $|E(M)|-|E(N)| \ge 5$.
