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

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