induction' n with k hk
a b c : ℤθ
n : ℕ
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (n + 1))).f = 1 + 3 ^ (n + 1) + 3 ^ (n + 2) * a ∧
      (↑(unit ^ 3 ^ (n + 1))).g = 3 ^ (n + 2) * b ∧ (↑(unit ^ 3 ^ (n + 1))).h = 3 ^ (n + 1) + 3 ^ (n + 2) * c
case zero
a b c : ℤθ
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.zero + 1))).f = 1 + 3 ^ (Nat.zero + 1) + 3 ^ (Nat.zero + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.zero + 1))).g = 3 ^ (Nat.zero + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.zero + 1))).h = 3 ^ (Nat.zero + 1) + 3 ^ (Nat.zero + 2) * c

case succ
a b c : ℤθ
k : ℕ
hk :
  ∃ a b c,
    (↑(unit ^ 3 ^ (k + 1))).f = 1 + 3 ^ (k + 1) + 3 ^ (k + 2) * a ∧
      (↑(unit ^ 3 ^ (k + 1))).g = 3 ^ (k + 2) * b ∧ (↑(unit ^ 3 ^ (k + 1))).h = 3 ^ (k + 1) + 3 ^ (k + 2) * c
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c