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

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