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

theorem
  (for n holds s.n = n/((n+1)!)) implies for n st n>=1 holds
  Partial_Sums(s).n = 1-1/((n+1)!)
proof
  defpred X[Nat] means Partial_Sums(s).$1= 1-1/(($1+1)!);
  assume
A1: for n holds s.n = n/((n+1)!);
  then
A2: s.0 = 0/((0+1)!) .= 0;
A3: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
A4: Partial_Sums(s).n =1-1/((n+1)!);
    n+2>=2 by NAT_1:11;
    then
A5: n+2>0 by XXREAL_0:2;
    Partial_Sums(s).(n+1)=1-1/((n+1)!)+ s.(n+1) by A4,SERIES_1:def 1
      .=1-1/((n+1)!)+(n+1)/((n+1+1)!) by A1
      .=1-(1*(n+2))/(((n+1)!)*(n+1+1))+(n+1)/((n+2)!) by A5,XCMPLX_1:91
      .=1-(1*(n+2))/((n+1+1)!)+(n+1)/((n+2)!) by NEWTON:15
      .=1-((n+2)/((n+1+1)!)-(n+1)/((n+2)!))
      .=1-((n+2)-(n+1))/((n+2)!) by XCMPLX_1:120
      .=1-1/((n+1+1)!);
    hence thesis;
  end;
  Partial_Sums(s).(1+0)=Partial_Sums(s).0+s.(1+0) by SERIES_1:def 1
    .= s.0 + s.1 by SERIES_1:def 1
    .= 1-1/((1+1)!) by A1,A2,NEWTON:14;
  then
A6: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A6,A3);
  hence thesis;
end;
