Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, $\#E(F_q)=q+1-t$ where $|t|\leq2\sqrt{q}$, and the structure of this group is either cyclic or almost cyclic.
Since the point addition and doubling formulas need multiplicative inverse to be computed so how can it be proven that all the elements in $E(F_q)$ have multiplicative inverse?