= Orientability of a regular fibre
{title2=$M\text{ oriented}\Longrightarrow f^{-1}(q)\text{ orientable}$}
Choose an <orientation of a vector space> on $T_qN$. The quotient identification for a <regular value> gives a smoothly oriented normal bundle along $F=f^{-1}(q)$. An oriented normal frame followed by a frame of $TF$ defines the latter's orientation through the ambient orientation. Changing normal lifts adds only tangent components, so the <determinant> sign is unchanged. No orientation on all of $N$ is required.
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