reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  (for n holds s.n>0 & s.n<1) implies for n holds (Partial_Product s).n>
  0 & (Partial_Product s).n<1
proof
  defpred X[Nat] means (Partial_Product s).$1>0 & (Partial_Product
  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>0 & (Partial_Product s).n<1;
    (Partial_Product s).(n+1)=((Partial_Product s).n)*(s.(n+1)) & s.(n+1)>
    0 by A1,Def1;
    hence thesis by A1,A3,XREAL_1:162;
  end;
  (Partial_Product s).0=s.0 by Def1;
  then
A4: X[0] by A1;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
