An elementary cancelling handle pair gives a product cobordism

The given configuration is exactly the hypothesis of [F1]: the handles $h^k$, $h^{k+1}$ are consecutive, the attaching sphere of $h^{k+1}$ meets the belt sphere of $h^k$ transversely in exactly one point, so $h^k,h^{k+1}$ form a geometrically cancelling pair.

Suppose $M_0$ is oriented and $1\le k\le n-2$, and use the induced orientations on the handles and middle level. Restrict to the connected collar component containing the prescribed boundary disc: its outgoing boundary before the pair is connected, as required by the matrix definition, and every other component has an empty handle list. This component’s two-handle presentation has a single $k$-handle and a single $(k+1)$-handle, so by the matrix definition the intersection matrix is the $1\times 1$ matrix with the single entry given by the oriented intersection number of the two spheres; by [F5] that entry is $\pm 1$ equal to the local intersection sign of the unique transverse point, and the finiteness underlying the count is [F6]. Without orientations [F5] supplies instead the mod-two entry $1$. Reorienting the core of $h^k$ or the cocore of $h^{k+1}$ reverses the induced orientation of the corresponding sphere and hence the sign of the entry alone, so the matrix is normalisable to $(1)$.

Conversely, by [F3] a geometrically cancelling consecutive pair with the standard complementary configuration may be inserted at any prescribed disc of the outgoing boundary of any presentation without changing the presented manifold relative to the incoming boundary; hence the displayed configuration is exactly the inverse of a trivial local modification of the collar.

Apply the cancellation theorem to the connected collar component containing the given outgoing disk, and leave all other collar components fixed. For the geometrically cancelling pair of step 1.1, the manifold $W=M_0\times [0,1]+h^k+h^{k+1}$ is diffeomorphic to the collar $M_0\times [0,1]$ relative to $M_0\times \{0\}$, and the diffeomorphism may be taken supported in a collar of the affected disc and the two handles; the cancelled manifold is presented relative to $M_0\times \{0\}$ by the empty handle list, which by [F4] is diffeomorphic to $M_0\times [0,\epsilon ]$, and the collar reparametrization gives $M_0\times [0,1]$.

The diffeomorphism $F:W\to M_0\times [0,1]$ of step 2.1 is the identity on $M_0\times \{0\}=M_0$ and carries the second face of $W$ onto $M_0\times \{1\}$, so it conjugates the inclusion $M_0\hookrightarrow W$ to the standard inclusion of the face $M_0\times \{0\}$ and the other face inclusion to the standard inclusion of $M_0\times \{1\}$ composed with a diffeomorphism of that face; both standard inclusions are homotopy equivalences (via the projections $(x,t)\mapsto (x,i)$ and the linear homotopies). Hence both face inclusions of $W$ are homotopy equivalences and, by [F7], $W$ is an h-cobordism.

Therefore an elementary cancelling pair attached to a collar builds a genuine product: $W\cong M_0\times [0,1]$ relative to $M_0\times \{0\}$, so $W$ is an h-cobordism, the connected component containing the pair has two-index intersection matrix $(\epsilon )$ with $\epsilon =\pm 1$ normalisable to $(1)$ when $M_0$ is oriented and $1\le k\le n-2$ (without orientations, in the same middle range, use the mod-two entry), and by step 1.3 the configuration is precisely the inverse of the trivial local modification that inserts a cancelling pair. This is the local model of the cancellation step of the h-cobordism theorem.