Cancellation Laws

Theorem
If $$G$$ is a group and $$a,b,c \in G,$$ then

$$ba=ca \Longrightarrow b=c$$ and $$ab=ac \Longrightarrow b=c$$.

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 $$ba=ca$$. Then,

$$(ba)a^{-1}=(ca)a^{-1}$$

$$\Longrightarrow b(aa^{-1})=c(aa^{-1})$$, by associativity

$$\Longrightarrow be=ce$$, by the definition of inverse

$$\Longrightarrow b=c$$, by the definition of identity

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

Alternatively, the fact that All Group Elements are Cancellable follows from the behaviour of invertible elements of a monoid.