reserve n for Nat,
  a,b for Real,
  s for Real_Sequence;

theorem
  (for n holds s.n = n |^ 3) implies for n holds Partial_Sums(s).n = n|^
  2*(n+1)|^2/4
proof
  defpred X[Nat] means Partial_Sums(s).$1 = $1|^2*($1+1)|^2/4;
  assume
A1: for n holds s.n = n |^ 3;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = n|^2*(n+1)|^2/4;
    then Partial_Sums(s).(n+1) = n|^2*(n+1)|^2/4 + s.(n+1) by SERIES_1:def 1
      .=n|^2*(n+1)|^2/4 + (n+1) |^ 3 by A1
      .=(n|^2*(n+1)|^2 + ((n+1) |^ 3)*4)/4
      .=(n|^2*(n+1)|^2 + (n+1)|^2*(n+1)*4)/4 by Lm2
      .=(n+1)|^2*(n|^2 +(2*2*n+4))/4
      .=(n+1)|^2*(n+2)|^2/4 by Lm3;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0 |^3 by A1
    .=0*(0+1)|^2/4 by NEWTON:11
    .=0|^2*(0+1)|^2/4 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
