 reserve n,s for Nat;

theorem
  (Triangle n) |^ 2 + (Triangle (n + 1)) |^ 2 = Triangle ((n + 1) |^ 2)
  proof
A1: Triangle (n + 1) = (n + 1) * (n + 1 + 1) / 2 by Th19
                    .= (n + 1) * (n + 2) / 2;
A2: (n + 1) |^ 2 = (n + 1) * (n + 1) by NEWTON:81;
    (Triangle n) |^ 2 + (Triangle (n + 1)) |^ 2 =
       (n * (n + 1) / 2) |^ 2 + (Triangle (n + 1)) |^ 2 by Th19
      .= (n * (n + 1) / 2) * (n * (n + 1) / 2) + ((n + 1) * (n + 2) / 2) |^ 2
          by NEWTON:81,A1
      .= (n * (n + 1) / 2) * (n * (n + 1) / 2) + ((n + 1) * (n + 2) / 2) *
          ((n + 1) * (n + 2) / 2) by NEWTON:81
      .= (n + 1) |^ 2 * ((n + 1) |^ 2 + 1) / 2 by A2
      .= Triangle (n + 1) |^ 2 by Th19;
    hence thesis;
  end;
