Identity of Monoid is Cancellable

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The identity of a monoid is cancellable.


Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$.

Let $x, y \in S$ such that $x \circ e = y \circ e$

Then, by the definition of the identity:

$x = x \circ e = y \circ e = y$

... thus $x = y$ and the result is proved.