Identity is Unique

Theorem: In a group $$G$$, the identity element is unique.

Proof
Let $$e$$ and $$e'$$ be identities of $$G$$. Then by the definition of identity, $$ge=g=e'g,$$ $$\forall g \in G$$. If we let $$g=e,$$ we have $$ee=e=e'e$$. If $$g=e',$$ we have $$e'e=e'=e'e'$$. From these two equalities we have $$e=e'e=e'$$. Thus, $$e=e'$$, proving there is only one identity in $$G$$.