cases b
case mk
b c : ℤθ
f✝ g✝ h✝ : ℤ
⊢ mul { f := f✝, g := g✝, h := h✝ } (add b c) =
    add (mul { f := f✝, g := g✝, h := h✝ } b) (mul { f := f✝, g := g✝, h := h✝ } c)
case mk.mk
c : ℤθ
f✝¹ g✝¹ h✝¹ f✝ g✝ h✝ : ℤ
⊢ mul { f := f✝¹, g := g✝¹, h := h✝¹ } (add { f := f✝, g := g✝, h := h✝ } c) =
    add (mul { f := f✝¹, g := g✝¹, h := h✝¹ } { f := f✝, g := g✝, h := h✝ }) (mul { f := f✝¹, g := g✝¹, h := h✝¹ } c)