Group Homomorphism of Product with Inverse

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

Then:


 * $(1): \quad \forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
 * $(2): \quad \forall x, y \in G: \map \phi {y^{-1} \circ x} = \paren {\map \phi y}^{-1} * \map \phi x$

Proof
Let $e_G$ be the identity of $G$ and $e_H$ be the identity of $H$.

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

Hence:
 * $x \circ y^{-1} \in \Dom \phi$

Result $(2)$:

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

Hence:
 * $y^{-1} \circ x \in \Dom \phi$