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

theorem
  (for n st n>=1 holds s.n = (2*n-1)|^2 & s.0 = 0) implies for n st n>=1
  holds Partial_Sums(s).n = n*(4*n|^2-1)/3
proof
  defpred X[Nat] means Partial_Sums(s).$1 = $1*(4*$1|^2-1)/3;
  assume
A1: for n st n>=1 holds s.n = (2*n-1)|^2 & 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*(4*n|^2-1)/3;
A4: n+1>=1 by NAT_1:11;
    Partial_Sums(s).(n+1) =n*(4*n|^2-1)/3 + s.(n+1) by A3,SERIES_1:def 1
      .=n*(4*n|^2-1)/3 + (2*(n+1)-1)|^2 by A1,A4
      .=(n*(4*n|^2-1) + (2*n+1)|^2*3)/3
      .=(n*(4*n|^2-1) + ((2*n)|^2+2*(2*n)*1+1|^2)*3)/3 by Lm3
      .=(n*4*n|^2-n*1 + ((2*n)|^2*3+2*(2*n)*3+1|^2*3))/3
      .=(n*(4*n|^2)-n + ((2|^2*n|^2)*3+2*2*n*3+1|^2*3))/3 by NEWTON:7
      .=(n*(4*n|^2)-n + ((2|^2*n|^2)*3+4*n*3+1*3))/3
      .=(n*(4*n|^2)-n + ((2*2*n|^2)*3+12*n+3))/3 by WSIERP_1:1
      .=(n*(4*n|^2) + (12-1)*n + 12*n|^2+3)/3
      .=(n+1)*(4*(n+1)|^2-1)/3 by Lm11;
    hence thesis;
  end;
  Partial_Sums(s).(0+1) = Partial_Sums(s).0 + s.(0+1) by SERIES_1:def 1
    .=s.0 + s.1 by SERIES_1:def 1
    .=0 + s.1 by A1
    .= 0 + (2*1-1)|^2 by A1
    .=1*(4*(1*1)-1)/3
    .= 1*(4*1|^2-1)/3;
  then
A5: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A5,A2);
  hence thesis;
end;
