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
Result $(1)$:

Result $(2)$: