Introduction to group theory/Right cancellation

Proof

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

Q.E.D.