Group Homomorphism of Product with Inverse

Theorem
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.

Then:

where $y^{-1}$ denotes the inverse of $y$.

Proof
Let $e_G$ and $e_H$ be the identities of $\struct {G, \circ}$ and $\struct {H, *}$ respectively.

By :
 * $\forall x, y \in G: x \circ y^{-1} \in G, y^{-1} \circ x \in G$


 * Result $(1):$:


 * Result $(2):$: