obtain ⟨k', hk'⟩ := hd
case succ.intro.intro.inl.intro
pn : ℕ
pk : ℕ+
hpk1 : (Nat.digits 10 (2 ^ pn.succ * ↑pk)).length = pn.succ
hpk2 : all_one_or_two (Nat.digits 10 (2 ^ pn.succ * ↑pk))
t : ℕ
ht : ↑pk = 2 * t
hd : 2 ^ pn.succ.succ ∣ prepend_two (2 ^ pn.succ * ↑pk)
⊢ ∃ k,
    (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k)).length = (pn + 1).succ ∧
      all_one_or_two (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k))
case succ.intro.intro.inl.intro.intro
pn : ℕ
pk : ℕ+
hpk1 : (Nat.digits 10 (2 ^ pn.succ * ↑pk)).length = pn.succ
hpk2 : all_one_or_two (Nat.digits 10 (2 ^ pn.succ * ↑pk))
t : ℕ
ht : ↑pk = 2 * t
k' : ℕ
hk' : prepend_two (2 ^ pn.succ * ↑pk) = 2 ^ pn.succ.succ * k'
⊢ ∃ k,
    (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k)).length = (pn + 1).succ ∧
      all_one_or_two (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k))