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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!

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    Can you add what you mean by "completely simple" to the question? Some may not be familiar with the definition. – 2017-02-07

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