Identity of Monoid is Cancellable

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Let $\struct {S, \circ}$ be a monoid.

Let $e$ be an identity element of $\struct {S, \circ}$.

Then $e$ is a cancellable element of $\struct {S, \circ}$.


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

Then, by definition of identity element:

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

Thus $x = y$.


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