Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$.

**Notation**: Let $F:U\to M$ be the inclusion map which is an embedding.
$F^*:\Omega^*(M)\to\Omega^*(U)$ be its pull-back. We denote $F^*\omega$ by $\omega|_U$.

Suppose $\omega_p =0$. Then $(\omega|_U)_p=(F^*\omega)_p=dF^*_p(\omega_{F(p)})=dF^*_p(\omega_{p})=0$, where the last equality follows by linearity of $dF^*_p$.

I can't prove the reverse implication. Any suggestion?