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