 reserve n,s for Nat;

theorem
  for n being non zero Nat holds
    3 * (Triangle n) + (Triangle (n -' 1)) = Triangle (2 * n)
  proof
    let n be non zero Nat;
A1: n -' 1 = n - 1 by XREAL_1:233,NAT_1:14;
A2: Triangle (n -' 1) = (n - 1) * (n - 1 + 1) / 2 by A1,Th19;
    3 * (Triangle n) + (Triangle (n -' 1)) =
      3 * (n * (n + 1) / 2) + (n - 1) * (n - 1 + 1) / 2 by A2,Th19
      .= 2 * n * (2 * n + 1) / 2
      .= Triangle (2 * n) by Th19;
    hence thesis;
  end;
