Introduction to group theory/Right cancellation


 * Proof

Let $$G$$ be a group and let $$a,b,c \in G$$ such that $$ac=bc$$. Since $$c \in G, \exist c^{-1} \in G$$ 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.