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

theorem
  (for n st n>=1 holds s.n = (2*n-1) |^ 3 & s.0 = 0) implies for n st n
  >=1 holds Partial_Sums(s).n = n|^2*(2*n|^2-1)
proof
  defpred X[Nat] means Partial_Sums(s).$1 = $1|^2*(2*$1|^2-1);
  assume
A1: for n st n>=1 holds s.n = (2*n-1) |^ 3 & s.0 = 0;
A2: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
A3: Partial_Sums(s).n = n|^2*(2*n|^2-1);
A4: n+1>=1 by NAT_1:11;
    Partial_Sums(s).(n+1) = n|^2*(2*n|^2-1) + s.(n+1) by A3,SERIES_1:def 1
      .=n|^2*(2*n|^2-1) + (2*(n+1)-1) |^ 3 by A1,A4
      .=n|^2*(2*n|^2-1) + (2*n+1) |^3
      .=n|^2*(2*n|^2-1) + ((2*n)|^3+3*(2*n)|^2+3*(2*n)+1) by Lm4
      .=n|^2*2*n|^2-n|^2 + (2*n)|^3+3*(2*n)|^2+3*2*n+1
      .=n|^2*2*n|^2-n|^2 + (2*n)|^3+3*(2|^2*n|^2)+6*n+1 by NEWTON:7
      .=n|^2*2*n|^2-n|^2 + 2|^3*n|^3+3*(2|^2*n|^2)+6*n+1 by NEWTON:7
      .=n|^2*2*n|^2-n|^2 + 2|^3*n|^3+3*(2*2*n|^2)+6*n+1 by WSIERP_1:1
      .=n|^2*2*n|^2-n|^2 + 2*2*2*n|^3+3*(2*2*n|^2)+6*n+1 by Lm1
      .=n|^2*n|^2*2+(12-1)*n|^2 + 8*n|^3+6*n+1
      .=(n+1)|^2*(2*(n+1)|^2-1) by Lm12;
    hence thesis;
  end;
  Partial_Sums(s).(1+0) =Partial_Sums(s).0 + s.(1+0) by SERIES_1:def 1
    .=s.0 + s.1 by SERIES_1:def 1
    .=s.1 + 0 by A1
    .= (2*1-1)|^3 by A1
    .=1|^(2+1)
    .=1|^2*(2*(1*1)-1)
    .=1|^2*(2*1|^2-1);
  then
A5: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A5,A2);
  hence thesis;
end;
