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.