have hind : (∀ n : N0, n ∈ A → (S n) ∈ A) := by
  intros n hel
  simp only [Set.mem_setOf_eq]
  simp only [Set.mem_setOf_eq] at hel
  rw [smul, hel, zero_plus_x_eq_eq]
x : ↑N0
A : Set ↑N0 := {x | mul (z, x) = z}
hzmem : z ∈ A
⊢ mul (z, x) = z
x : ↑N0
A : Set ↑N0 := {x | mul (z, x) = z}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ mul (z, x) = z