I am studying semigroups. I saw a Lemma in the text that states:

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> Let $e$ be an idempotent of the monoid $M$, $x$, $y$ be two elements of $eMe$. Then, $(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$ if and only if $MxM = MyM$.
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If we have $MxM=MyM$, then by $x = exe$ and $y = eye$, it is not hard. I can't figure the other direction, but the text states the other direction is trivial.