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

theorem
  (for n holds s.n = n*(n+1)*2|^n) implies for n holds Partial_Sums(s).n
  = 2|^(n+1)*(n|^2-n+2)-4
proof
  defpred X[Nat] means Partial_Sums(s).$1=2|^($1+1)*($1|^2-$1+2)-4;
  assume
A1: for n holds s.n = n*(n+1)*2|^n;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = 2|^(n+1)*(n|^2-n+2)-4;
    then Partial_Sums(s).(n+1) =2|^(n+1)*(n|^2-n+2)-4 + s.(n+1) by
SERIES_1:def 1
      .=2|^(n+1)*(n|^2-n+2)-4 + (n+1)*(n+1+1)*2|^(n+1) by A1
      .=2|^(n+1)*2*((n|^2-n+2) + (n+1)*(n+2))/2-4
      .=2|^(n+1+1)*((n|^2-n+2) + (n+1)*(n+2))/2-4 by NEWTON:6
      .=2|^(n+2)*(n|^2-n+2 + n*n+1*n+n*2+2)/2-4
      .=2|^(n+2)*(n|^2-n+2 + n|^2+1*n+n*2+2)/2-4 by WSIERP_1:1
      .=2|^(n+2)*(n|^2 + 2*n +1-1- n + 2)-4
      .=2|^(n+2)*(n|^2 + 2*n*1 +1|^2-1- n + 2)-4
      .=2|^(n+2)*((n+1)|^2-1- n + 2)-4 by Lm3
      .=2|^(n+2)*((n+1)|^2-(1+ n) + 2)-4;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .= 0*(0+1)*2|^0 by A1
    .=2*2-4
    .=2|^(0+1)*(0-0+2)-4
    .=2|^(0+1)*(0|^2-0+2)-4 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
