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