simp only [mul, add]
case mk.mk.mk
z✝² w✝² z✝¹ w✝¹ z✝ w✝ : ℤ
⊢ mul (add { z := z✝², w := w✝² } { z := z✝¹, w := w✝¹ }) { z := z✝, w := w✝ } =
    add (mul { z := z✝², w := w✝² } { z := z✝, w := w✝ }) (mul { z := z✝¹, w := w✝¹ } { z := z✝, w := w✝ })
case mk.mk.mk
z✝² w✝² z✝¹ w✝¹ z✝ w✝ : ℤ
⊢ { z := (z✝² + z✝¹) * z✝ - 2 * (w✝² + w✝¹) * w✝, w := (z✝² + z✝¹) * w✝ + (w✝² + w✝¹) * z✝ + (w✝² + w✝¹) * w✝ } =
    { z := z✝² * z✝ - 2 * w✝² * w✝ + (z✝¹ * z✝ - 2 * w✝¹ * w✝),
      w := z✝² * w✝ + w✝² * z✝ + w✝² * w✝ + (z✝¹ * w✝ + w✝¹ * z✝ + w✝¹ * w✝) }