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

theorem
  (for n holds s.n = n!*n) implies for n st n>=1 holds Partial_Sums(s).n
  = (n+1)!-1
proof
  defpred X[Nat] means Partial_Sums(s).$1= ($1+1)!-1;
  assume
A1: for n holds s.n = n!*n;
  then
A2: s.0 = 0!*0 .= 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 =(n+1)!-1;
    Partial_Sums(s).(n+1)=(n+1)!-1+ s.(n+1) by A4,SERIES_1:def 1
      .=(n+1)!-1 +(n+1)!*(n+1) by A1
      .=(n+1)!*(n+1+1)-1
      .=(n+1+1)!-1 by NEWTON:15;
    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 by A1,A2
    .= (1+1)!-1 by NEWTON:13,14;
  then
A5: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A5,A3);
  hence thesis;
end;
