suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from
  Finset.prod_congr rfl this
m : ℕ → ℕ+
k : ℕ
nm : ℕ+
⊢ ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(if i < k then m i else nm)) = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))
m : ℕ → ℕ+
k : ℕ
nm : ℕ+
⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)