Vector field is normal to image moved by a flow

Question

Assume that

1. $X$ is a vector field on a domain in $\mathbb{R}^n$,
2. $F$ is its flow i.e. $\frac{\partial }{\partial t}F(t,x) = X_{F(t,x)}$, and
3. $D$ is $n-1$ dimensional compact surface s.t. $X$ is normal to $D$.

Then there is $\epsilon>0$ s.t. $X$ is still normal to $F(t,D)$ for all $0<t<\epsilon$.

Proof : Define $f$ to be a value $t$ on $F(t,D)$. Hence ${\rm grad}\ f$ is normal to $F(t,D)$.

If $c(t)=F(t,x)$, then $$1=\frac{d}{dt}f\circ c(t)= {\rm grad}\ f \cdot X_{F(t,x)}$$

How can we finish the proof ?