$N$-detachable pairs in 3-connected matroids III: the theorem

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 $|E(M)|-|E(N)| \ge 5$.