Let $\pi:E\to M$ be a smooth fibre bundle with local descriptions over an open cover $(U_i)_{i\in I}$ and transition functions $g_{ij}:U_i\cap U_j\to \operatorname{Diff}(F)$. On all nonempty overlaps, the following identities hold:
These hold for every $x\in U_i\cap U_j\cap U_k$. For a smooth rank $k$ vector bundle with smooth vector-bundle trivializations, the same identities hold with composition replaced by matrix multiplication in $GL(k,\mathbb R)$.