cases c
case mk.mk
c : ℤα
z✝¹ w✝¹ z✝ w✝ : ℤ
⊢ mul { z := z✝¹, w := w✝¹ } (add { z := z✝, w := w✝ } c) =
    add (mul { z := z✝¹, w := w✝¹ } { z := z✝, w := w✝ }) (mul { z := z✝¹, w := w✝¹ } c)
case mk.mk.mk
z✝² w✝² z✝¹ w✝¹ z✝ w✝ : ℤ
⊢ mul { z := z✝², w := w✝² } (add { z := z✝¹, w := w✝¹ } { z := z✝, w := w✝ }) =
    add (mul { z := z✝², w := w✝² } { z := z✝¹, w := w✝¹ }) (mul { z := z✝², w := w✝² } { z := z✝, w := w✝ })