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

theorem
  (for n holds s.n = n|^5) implies for n holds Partial_Sums(s).n = n|^2*
  (n+1)|^2*(2*n|^2+2*n-1)/12
proof
  defpred X[Nat] means Partial_Sums(s).$1=$1|^2*($1+1)|^2*(2*$1|^2+
  2*$1-1)/12;
  assume
A1: for n holds s.n = n|^5;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = n|^2*(n+1)|^2*(2*n|^2+2*n-1)/12;
    then Partial_Sums(s).(n+1) =n|^2*(n+1)|^2*(2*n|^2+2*n-1)/12 + s.(n+1) by
SERIES_1:def 1
      .=n|^2*(n+1)|^2*(2*n|^2+2*n-1)/12 + (n+1)|^5 by A1
      .=(n|^2*(n+1)|^2*(2*n|^2+2*n-1)+(n+1)|^(3+2)*12)/12
      .=(n|^2*(n+1)|^2*(2*n|^2+2*n-1)+(n+1)|^3*(n+1)|^2*12)/12 by NEWTON:8
      .=((n+1)|^2*(n|^2*(2*n|^2+2*n-1)+(n+1)|^3*12))/12
      .=((n+1)|^2*((n+2)|^2*(2*(n+1)|^2+2*(n+1)-1)))/12 by Lm15
      .=(n+1)|^2*(n+2)|^2*(2*(n+1)|^2+2*(n+1)-1)/12;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0 |^5 by A1
    .=0*(0+1)|^2*(2*0|^2+2*0-1)/12 by NEWTON:11
    .= 0|^2*(0+1)|^2*(2*0|^2+2*0-1)/12 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
