 reserve n,s for Nat;

theorem
  (Triangle n) |^ 2 = Sum NPower (3, n)
  proof
    defpred P[Nat] means
      (Triangle $1)|^2 = Sum NPower (3, $1);
A1: P[0]
    proof
      thus (Triangle 0) |^ 2 = 0 * 0 by NEWTON:81
         .= Sum NPower (3, 0);
    end;
A2: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A3:   P[k];
      reconsider k1 = k + 1 as Element of REAL by XREAL_0:def 1;
      (Triangle (k + 1)) |^ 2 = ((k + 1) * (k + 1 + 1) / 2) |^ 2 by Th19
          .= ((k + 1) * (k + 2) / 2) * ((k + 1) * (k + 2) / 2) by NEWTON:81
          .= (k + 1) * (k + 1) * (k + 2) * (k + 2) / (2 * 2)
          .= (k + 1) |^ 2 * (k + 2) * (k + 2) / (2 * 2) by NEWTON:81
          .= (k + 1) |^ 2 * (k * k + 4 * k + 4) / (2 * 2)
          .= (k + 1) |^ 2 * (k |^ 2 + 4 * k + 4) / (2 * 2) by NEWTON:81
          .= ((k + 1) |^ 2 * k |^ 2) / 4 + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4
          .= (((k + 1) * k) |^ 2) / (2 * 2) + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by NEWTON:7
          .= (((k + 1) * k) |^ 2) / (2 |^ 2) + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by NEWTON:81
          .= ((k + 1) |^ 2 * k |^ 2) / (2 |^ 2) + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by NEWTON:7
          .= (k + 1) * (k + 1) * k |^ 2 / (2 |^ 2) + ((k + 1) |^ 2 * 4 * k)
               / 4 + ((k + 1) |^ 2 * 4) / 4 by NEWTON:81
          .= (k + 1) * (k + 1) * (k * k) / (2 |^ 2) + ((k + 1) |^ 2 * 4 * k)
               / 4 + ((k + 1) |^ 2 * 4) / 4 by NEWTON:81
          .= (k + 1) * (k + 1) * k * k / (2 * 2) + ((k + 1) |^ 2 * 4 * k) / 4
               + ((k + 1) |^ 2 * 4) / 4 by NEWTON:81
          .= ((k + 1) * k / 2) * ((k + 1) * k / 2) + ((k + 1) |^ 2 * 4 * k) /
               4 + ((k + 1) |^ 2 * 4) / 4
          .= (Triangle k) * ((k + 1) * k / 2) + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by Th19
          .= (Triangle k) * (Triangle k) + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by Th19
          .= (Triangle k) |^ 2 + ((k + 1) |^ 2 * 4 * k) / 4 +
               ((k + 1) |^ 2 * 4) / 4 by NEWTON:81
          .= Sum NPower (3, k) + (k + 1) |^2 * (k + 1) by A3
          .= Sum NPower (3, k) + ((k + 1) * (k + 1) * (k + 1)) by NEWTON:81
          .= Sum NPower (3, k) + ((k + 1) |^ 3) by POLYEQ_3:27
          .= Sum (NPower (3, k) ^ <* k1 |^ 3 *>) by RVSUM_1:74
          .= Sum NPower (3, k + 1) by Th58;
      hence thesis;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
