Cancellation Laws

Theorem
Let $$G$$ be a group.

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

Then:
 * $$b a = c a \implies b = c$$;
 * $$a b = a c \implies b = c$$.

These are respectively called the right and left cancellation laws.

Proof
Let $$a, b, c \in G$$ and let $$a^{-1}$$ be the inverse of $$a$$.

Suppose $$b a = c a$$.

Then:

$$ $$ $$ $$

Thus, the right cancellation law holds. The proof of the left cancellation law is analogous.

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

From Invertible also Cancellable, it follows that all its elements are therefore cancellable.