 reserve n,s for Nat;

theorem
  for n being non zero Nat holds
    Triangle n + Triangle (n -' 1) = (1 + 2 * n - 1) * n / 2
  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;
    Triangle n + Triangle (n -' 1)
      = n * (n + 1) / 2 + Triangle (n-'1) by Th19
     .= (1 + 2 * n - 1) * n / 2 by A2;
    hence thesis;
  end;
