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

theorem
  (for n holds s.n = n|^7) implies for n holds Partial_Sums(s).n = n|^2*
  (n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)/24
proof
  defpred X[Nat] means Partial_Sums(s).$1=$1|^2*($1+1)|^2*(3*$1|^4+
  6*$1|^3-$1|^2-4*$1+2)/24;
  assume
A1: for n holds s.n = n|^7;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n =n|^2*(n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)/24;
    then
    Partial_Sums(s).(n+1) =n|^2*(n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)/24+ s.
    (n+1) by SERIES_1:def 1
      .=n|^2*(n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)/24+ (n+1)|^7 by A1
      .=(n|^2*(n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)+ (n+1)|^(5+2)*24)/24
      .=(n|^2*(n+1)|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)+ (n+1)|^5*(n+1)|^2*24)/24
    by NEWTON:8
      .=(n+1)|^2*(n|^2*(3*n|^4+6*n|^3-n|^2-4*n+2)+ (n+1)|^5*24)/24
      .=(n+1)|^2*((n+2)|^2*(3*(n+1)|^4+6*(n+1)|^3-(n+1)|^2-4*(n+1)+2))/24 by
Lm17
      .=(n+1)|^2*(n+1+1)|^2*(3*(n+1)|^4+6*(n+1)|^3-(n+1)|^2-4*(n+1)+2)/24;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0 |^7 by A1
    .= 0*(0+1)|^2*(3*0|^4+6*0|^3-0|^2-4*0+2)/24 by NEWTON:11
    .= 0|^2*(0+1)|^2*(3*0|^4+6*0|^3-0|^2-4*0+2)/24 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
