Group Homomorphism of Product with Inverse

Theorem
Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism. Then:


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

Proof

 * First result:


 * Second result: