Group Homomorphism Preserves Inverses/Proof 3

Proof
From Group Homomorphism of Product with Inverse, we have:
 * $\forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$

Putting $x = e_G$ and $y = x$ we have: