 Quotient vector bundle

For a smooth vector subbundle $F\subset E$, the quotient has fibers $(E/F{)}_p=E_p/F_p$. Extend a local frame of a vector bundle of $F$ to one of $E$; the classes of the complementary frame vectors trivialize the quotient. Adapted transition functions of a vector bundle are block upper triangular, and their lower-right blocks give the quotient transitions. These are smooth and satisfy the cocycle identity, defining a smooth vector bundle. The projection $E\to E/F$ is a smooth vector bundle morphism.