Introduction to group theory/Right cancellation

Proof

Let $G$ be a group and let $a,b,c\in G$ such that $ac=bc$. Since $c\in G,\mathrm{\exists }{c}^{-1}\in G$ such that $c*c^{-1}=e$. Multiplying $ac=bc$ on each side by $c^{-1}$ we obtain. $ac*c^{-1}=bc*c^{-1}$. Applying the definition of inverses ($c*c^{-1}=e$) we get $a*e=b*e$ Applying the definition of identity we get $a=b$

Q.E.D.