apply Or.inl ∘ Finset.mem_fold_union_iff.mpr
case zero.succ
n : ℕ
stmt : Variable → Variable → ℕ → LoopProgram
h_stmt : stmt = add ∨ stmt = sub
y' : ℕ
⊢ stmt { index := Nat.zero } { index := Nat.succ y' } n ∈
      Finset.fold (fun a b => a ∪ b) ∅ extendLenOne (ofLen (Nat.zero + Nat.succ y' + n)) ∨
    stmt { index := Nat.zero } { index := Nat.succ y' } n ∈
        Finset.fold (fun x x_1 => x ∪ x_1) ∅
          (fun x =>
            Finset.image (fun x => concat x.1 x.2)
              (Finset.product (ofLen ↑x) (ofLen (Nat.zero + Nat.succ y' + n - ↑x))))
          (Finset.attach (Finset.range (Nat.zero + Nat.succ y' + n))) ∨
      stmt { index := Nat.zero } { index := Nat.succ y' } n ∈ baseProgram (Nat.zero + Nat.succ y' + n + 1)
case zero.succ
n : ℕ
stmt : Variable → Variable → ℕ → LoopProgram
h_stmt : stmt = add ∨ stmt = sub
y' : ℕ
⊢ ∃ x ∈ ofLen (Nat.zero + Nat.succ y' + n), stmt { index := Nat.zero } { index := Nat.succ y' } n ∈ extendLenOne x