 reserve n,s for Nat;

theorem
  for n being non trivial Nat holds
    Triangle n + Triangle (n -' 1) * Triangle (n + 1) = (Triangle n) |^ 2
  proof
    let n be non trivial Nat;
A1:Triangle n = n * (n + 1) / 2 by Th19;
    0 + 1 <= n by NAT_1:13; then
A2: n -' 1 = n - 1 by XREAL_1:233;
A3: Triangle (n -' 1) = (n -' 1) * (n -' 1 + 1) / 2 by Th19
      .= (n - 1) * n / 2 by A2;
    Triangle (n + 1) = (n + 1) * (n + 1 + 1) / 2 by Th19
                    .= (n + 1) * (n + 2) / 2;then
    Triangle n + Triangle (n -' 1) * Triangle (n + 1) =
        Triangle n * Triangle n by A1,A3
     .= (Triangle n) |^ 2 by NEWTON:81;
    hence thesis;
  end;
