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**Also sprach Zarathustra** Hello!

I have proved the following lemma:

If $\displaystyle p$ is a prime number and $\displaystyle A,B$ are subsets of $\displaystyle \mathbb{Z}_p$ and $\displaystyle \emptyset \neq A \neq \mathbb{Z}_p$, $\displaystyle |B|=2$, then $\displaystyle |A+B| \geq |A|+1$ when $\displaystyle A+B=\{a+b|a \in A, b \in B \}$.

With that lemma I'm trying to prove the following:

If $\displaystyle 0 \leq a_1 \leq a_2 \leq...\leq a_{2p-1}<p $ and if $\displaystyle a_i \neq a_{i+p-1}$ for all $\displaystyle 1<i \leq p$. Let us now define $\displaystyle A_1=\{a_1\}$, $\displaystyle A_i=\{a_i,a_{i+p-1}\}$ for $\displaystyle 1<i \leq p$.

By repeating process with the lemma I need to conclude:

$\displaystyle |A_2+A_3+...+A_p|=p$

But I have some difficulties proving that... I could have proved only the inequality:

$\displaystyle |A_2+A_3+...+A_p|\geq p$

Please help.