cases' hk with p hp
case succ
a b c : ℤθ
k : ℕ
hk :
  ∃ a b c,
    (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * a ∧
      (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * b ∧
        (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * c
l : k + 1 = Nat.succ k
⊢ ∃ 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
case succ.intro
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p : ℤ
hp :
  ∃ b c,
    (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p ∧
      (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * b ∧
        (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * 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