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Local defining functions $f_{\alpha }$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_{\alpha }/f_{\beta }$ on overlaps. These ratios define a real line bundle with a global transverse section whose local representatives are the $f_{\alpha }$ and whose zero set is $Y$. The bundle is naturally isomorphic to the normal bundle of $Y$.

Choose a locally finite cover $(U_{\alpha })$ such that on every chart meeting $Y$ there is a smooth defining function $f_{\alpha }$ withand take $f_{\alpha }=1$ on charts disjoint from $Y$. On an overlap, the supplied division lemma extendssmoothly across $Y$. After shrinking the charts, this extension is nowhere zero. The identities $g_{\alpha \beta }{g}_{\beta \gamma }=g_{\alpha \gamma }$ make these functions transition functions for a real line bundle $L\to X$.

Choose local frames $e_{\alpha }$ with $e_{\beta }=g_{\alpha \beta }{e}_{\alpha }$. Then the local sectionsagree on overlaps and define a global section. Its zero set is exactly $Y$. Along $Y$, its vertical derivative is represented by the nonzero covector $d{f}_{\alpha }$, so $s$ is transverse to the zero section, as in the transverse intersection theorem. This is the defining line bundle of a properly embedded hypersurface.