theorem Th43:
  (for n holds s.n>0) implies (Partial_Product s).n>0
proof
  defpred X[Nat] means (Partial_Product s).$1>0;
  assume
A1: for n holds s.n>0;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A3: (Partial_Product s).n>0;
    (Partial_Product s).(n+1) = (Partial_Product s).n*s.(n+1) & s.(n+1)>0
    by A1,Def1;
    hence thesis by A3;
  end;
  s.0>0 by A1;
  then
A4: X[0] by Def1;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
