Right Cancellable Commutative Operation is Left Cancellable
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be right cancellable and also commutative.
Then $\circ$ is also left cancellable.
Proof
Let $\circ$ be both right cancellable and commutative on a set $S$.
Then:
\(\ds a \circ b\) | \(=\) | \(\ds a \circ c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds b \circ a\) | \(=\) | \(\ds c \circ a\) | $\circ$ is Commutative | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds c\) | $\circ$ is Right Cancellable |
$\blacksquare$