Cancellation Laws/Proof 2

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Let $G$ be a group.

Let $a, b, c \in G$.

Then the following hold:

Right cancellation law
$b a = c a \implies b = c$
Left cancellation law
$a b = a c \implies b = c$


From its definition, a group is a monoid, all of whose elements have inverses and thus are invertible.

From Invertible Element of Monoid is Cancellable, it follows that all its elements are therefore cancellable.