reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  (for n holds s.n>0 & s.n<1) implies for n holds Partial_Product(s).n>=
  Partial_Sums(s).n-n
proof
  defpred X[Nat] means Partial_Product(s).$1>=Partial_Sums(s).$1-$1;
  assume
A1: for n holds s.n>0 & s.n<1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A3: Partial_Product(s).n>=Partial_Sums(s).n-n;
A4: s.(n+1)<1 by A1;
    Partial_Sums(s).n-(n+1)<0 by A1,Th50,XREAL_1:49;
    then s.(n+1)*(Partial_Sums(s).n-n-1)>1*(Partial_Sums(s).n-n-1) by A4,
XREAL_1:69;
    then
A5: s.(n+1)*Partial_Sums(s).n-s.(n+1)*n-s.(n+1)+s.(n+1)> Partial_Sums(s).
    n-n-1+s.(n+1) by XREAL_1:8;
    s.(n+1)>0 by A1;
    then
Partial_Product(s).n*s.(n+1)>=(Partial_Sums(s).n-n)*s.(n+1) by A3,XREAL_1:64;
    then Partial_Product(s).n*s.(n+1)>Partial_Sums(s).n+s.(n+1)-(n+1) by A5,
XXREAL_0:2;
    then Partial_Product(s).(n+1)>Partial_Sums(s).n+s.(n+1)-(n+1) by
SERIES_3:def 1;
    hence thesis by SERIES_1:def 1;
  end;
  Partial_Sums(s).0-0 = s.0 by SERIES_1:def 1;
  then
A6: X[0] by SERIES_3:def 1;
  for n holds X[n] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
