On distinct cross-ratios in positive characteristic

We prove that a finite subset $A$ of the projective line $F{P}^1$ over a field $F$ of positive characteristic $p$ determines $\mathrm{\Omega }(\mathrm{\mid }A{\mathrm{\mid }}^2)$ distinct cross-ratios, as long as $\mathrm{\mid }A\mathrm{\mid }<\sqrt{p}\mathrm{.}$