 reserve n,s for Nat;

theorem
  for n being non zero Nat holds
    Triangle (n -' 1) + 6 * (Triangle n) + Triangle (n + 1) =
      8 * (Triangle n) + 1
  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;
A3: Triangle n = n * (n + 1) / 2 by Th19;
    Triangle (n + 1) = (n + 1) * (n + 1 + 1) / 2 by Th19
      .= (n + 1) * (n + 2) / 2;
    hence thesis by A3,A2;
  end;
