= Associativity of twisted half-smash products
For isometry families $f_x:U\to U\prime$ and $g_y:U\prime\to U\prime\prime$, the complement of $g_yf_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same <Thom space> as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y\ltimes_g(X\ltimes_fE)\cong(Y\times X)\ltimes_{g\circ f}E$ in the <stable homotopy category>.
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