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, splus, zplus]
x : ↑N0
A : Set ↑N0 := {x | mul (S z, x) = x}
hzmem : z ∈ A
⊢ mul (S z, x) = x
x : ↑N0
A : Set ↑N0 := {x | mul (S z, x) = x}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ mul (S z, x) = x