induction' n with n ih
x y n : ℕ
⊢ (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
case zero
x y : ℕ
⊢ (x + y) ^ Nat.zero = ∑ k in Finset.range (Nat.zero + 1), Nat.choose Nat.zero k * x ^ (Nat.zero - k) * y ^ k

case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ Nat.succ n = ∑ k in Finset.range (Nat.succ n + 1), Nat.choose (Nat.succ n) k * x ^ (Nat.succ n - k) * y ^ k