Handle slides preserve the relative diffeomorphism type

Since the original attaching regions are disjoint, their quotient gluings commute: attach $h_2$ first and regard $h_1$ as attached to $N={\mathrm{\partial }}_{+}(W\cup h_2)$. This reordering fixes the lower stage. A framed parallel of the second attaching sphere bounds the parallel core disk $D^k\times \{z\}$ in the outgoing region of $h_2$, for a chosen $z\in S^{n-k-1}$.

Extend the slide band into that parallel core disk. A collar of the band and disk provides an embedded strip $\mathrm{\Phi }:[0,2]\times D^{k-1}\to N$, starting with a disk on the first attaching sphere. Choose a smooth function $b$ on $D^{k-1}$, equal to one on a smaller disk and zero on a neighbourhood of its boundary. Replace the first attaching-sphere disk $\mathrm{\Phi }(0,y)$ by $\mathrm{\Phi }(2tb(y),y)$, leaving the rest of the sphere fixed. This is an isotopy: each moving patch is a graph in the strip, and the cutoff makes it match the fixed patch smoothly. Its image misses the fixed rest after the strip is chosen thin. At the final time the patch has traversed the parallel core; the final sphere can be chosen disjoint from the belt sphere of $h_2$. Shrink its normal disk neighbourhood to retain that disjointness. A radial diffeotopy in $D^k\times S^{n-k-1}$, fixed on a smaller belt neighbourhood and extended across the seam collar, then carries this compact attaching region back into the old-boundary summand. This gives the band sum with the framed parallel sphere, with the framing transported by the same diffeotopy. In the case $k=1$ this moves one attaching point across the interval core to a parallel of its other endpoint. This is Wall's disk push, Handle Addition Theorem, printed p. 148.

The normal framing travels in the strip coordinates. Shrink the normal disk radius uniformly along the compact isotopy; its normal thickening is then an isotopy of the full attaching-region embeddings, with the band-sum framing at the endpoint. Smoothly reparametrize it to be stationary near the time endpoints. It takes place in $N$, after $h_2$ has been attached; a slide need not be an isotopy in the original boundary $M$.

Apply [F2] to the base $W\cup h_2$ and the isotopy in step 3.1. The resulting attachments of $h_1$ and $h_1^{\mathrm{\prime }}$ are diffeomorphic relative to the incoming boundary, supported near the swept strip and the two handles, and later attaching data are transported by this diffeomorphism. Reorder the disjoint original attachments as in step 1.1 to obtain the asserted comparison.