simp [A, Set.ext_iff] at heq
x y : ↑N0
A : Set ↑N0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ plus (S x, y) = S (plus (x, y))
x y : ↑N0
A : Set ↑N0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
  ∀ (x_1 : α),
    (∃ (x_2 : x_1 ∈ N0),
        plus (S x, { val := x_1, property := (_ : x_1 ∈ N0) }) =
          S (plus (x, { val := x_1, property := (_ : x_1 ∈ N0) }))) ↔
      x_1 ∈ N0
⊢ plus (S x, y) = S (plus (x, y))