One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$

Let $A$ and $B$ be sets of $k\ge 5$ elements in $F=\mathbb{Z}\mathrm{/}p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$, $a\mathrm{\ne }b$. The number of elements of this set is at least $2k-3$. Károlyi showed that, except from some particular cases, The equality can only occur if $A=B$ and $A$ is an arithmetic progression with non zero difference. We prove that in the case that $\mathrm{\mid }A\dot{+}B\mathrm{\mid }=2k-2$ and $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }$ the equality $A=B$ holds.