cases 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)
case mk.mk.mk
x✝² y✝² x✝¹ y✝¹ x✝ y✝ : ℤ
⊢ mul (add { x := x✝², y := y✝² } { x := x✝¹, y := y✝¹ }) { x := x✝, y := y✝ } =
    add (mul { x := x✝², y := y✝² } { x := x✝, y := y✝ }) (mul { x := x✝¹, y := y✝¹ } { x := x✝, y := y✝ })