Is there a simple proof of the following fact:
Fact. Let $S$ be a completely simple semigroup with cancellations, i.e. each of the equalities $xa=xb$, $ax=bx$ implies $a=b$. Prove that $S$ is a group.
Using Sushkevich-Rees Theorem, I can prove it, but my proof is not elegant.
Can you prove this fact using only simple arguments? Probably, you know a paper, where this result was proved at the first time?
Thank you for your help!