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

theorem
  (for n holds s.n = 1/(n+1)) implies (Partial_Product s).n = 1/((n+1)!)
proof
  defpred X[Nat] means (Partial_Product s).$1 = 1/(($1+1)!);
  assume
A1: for n holds s.n = 1/(n+1);
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume (Partial_Product s).n = 1/((n+1)!);
    then (Partial_Product s).(n+1) = 1/((n+1)!)* s.(n+1) by SERIES_3:def 1
      .=1/((n+1)!)*(1/(n+1+1)) by A1
      .=(1/((n+1)!)*1)/(n+1+1) by XCMPLX_1:74
      .= 1/((n+1)!*((n+1)+1)) by XCMPLX_1:78
      .= 1/((n+2)!) by NEWTON:15;
    hence thesis;
  end;
  (Partial_Product s).0 = s.0 by SERIES_3:def 1
    .= 1/((0+1)!) by A1,NEWTON:13;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
