theorem
  (for n holds s.n>=1) implies for n holds (Partial_Product s).n>=1
proof
  defpred X[Nat] means (Partial_Product s).$1>=1;
  assume
A1: for n holds s.n>=1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
A3: (Partial_Product s).(n+1)=((Partial_Product s).n)*(s.(n+1)) by Def1;
    assume
A4: (Partial_Product s).n>=1;
    then ((Partial_Product s).n)<=((Partial_Product s).n)*(s.(n+1)) by A1,
XREAL_1:151;
    hence thesis by A4,A3,XXREAL_0:2;
  end;
  (Partial_Product s).0=s.0 by Def1;
  then
A5: X[0] by A1;
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
