cases b
case mk
b c : ℤω
x✝ y✝ : ℤ
⊢ mul (add { x := x✝, y := y✝ } b) c = add (mul { x := x✝, y := y✝ } c) (mul b c)
case mk.mk
c : ℤω
x✝¹ y✝¹ x✝ y✝ : ℤ
⊢ mul (add { x := x✝¹, y := y✝¹ } { x := x✝, y := y✝ }) c =
    add (mul { x := x✝¹, y := y✝¹ } c) (mul { x := x✝, y := y✝ } c)