 reserve n,s for Nat;

theorem
  (Triangle (n + 1)) |^ 2 - (Triangle n) |^ 2 = (n + 1) |^ 3
  proof
A1:  Triangle (n + 1) = (n + 1) * (n + 1 + 1) / 2 by Th19
                     .= (n + 1) * (n + 2) / 2;
A2:  (n + 1) |^ 3 = (n + 1) * (n + 1) * (n + 1) by POLYEQ_3:27
        .= n * n * n + 3 * n * n + 3 * n + 1;
A3:  (n + 1) * (n + 1) = (n + 1) |^ 2 by NEWTON:81;
     (Triangle (n + 1)) |^ 2 - (Triangle n) |^ 2 =
        ((n + 1) * (n + 2) / 2) |^ 2 - (n * (n + 1) / 2) |^ 2 by Th19,A1
     .= ((n + 1) * (n + 2) / 2) * ((n + 1) * (n + 2) / 2) -
        (n * (n + 1) / 2) |^ 2 by NEWTON:81
     .= (n + 1) * (n + 1) * (n + 2) * (n + 2) / 4 - (n * (n + 1) / 2) |^ 2
     .= (n + 1) |^ 2 * (n + 2) * (n + 2) / 4 - (n * (n + 1) / 2) |^ 2
           by NEWTON:81
     .= (n + 1) |^ 2 * (n + 2) * (n + 2) / 4 - (n * (n + 1) / 2) *
         (n * (n + 1) / 2) by NEWTON:81
     .= (n + 1) |^ 2 * (n + 2) * (n + 2) / 4 - n * n * (n + 1) * (n + 1) / 4
     .= (n + 1) |^ 2 * (n + 2) * (n + 2) / 4 - n |^ 2 * (n + 1) * (n + 1) / 4
           by NEWTON:81
     .= (n + 1) |^ 2 * (n * n + 2 * n + 2 * n + 4 - n |^ 2) / 4 by A3
     .= (n + 1) |^ 2 * (n |^ 2 + 2 * n + 2 * n + 4 - n |^ 2) / 4 by NEWTON:81
     .= (n + 1) |^ 2 * (n + 1)
     .= (n + 1) * (n + 1) * (n + 1) by NEWTON:81
     .= (n + 1) |^ 3 by A2;
    hence thesis;
  end;
