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