Let P be the projective plane obtained by identifying antipodal points on the unit sphere.
Let $\alpha$ be a curve in P.
The book I am reading says that $\alpha'(t)$ is the function such that
for every differentiable real valued function $f$ on P.
Let $F$ be the projection of the unit sphere to P and $F*$ the tangent map.
How to prove that two distinct tangent vectors to the unit sphere with the same point of application result in distinct tangent vectors to the projective plane with the same point of application under the tangent map? In other words, how to prove
If $v\neq w$, $v ,w$ are both tangent vectors to the unit sphere at point say $q$. Then $F*(v)\neq F*(w)$? How to write detailed proof?