
theorem Th34:
  for a be NAT-valued Real_Sequence, b, c be Nat
  st b >= 2 & c >= 1 & rng a c= c & c <= b holds
  for i be Nat holds Liouville_seq (a,b).i <= ((c - 1)(#)(powerfact b)).i
  proof
    let a be NAT-valued Real_Sequence,b, c be Nat;
    assume
A1: b >= 2 & c >= 1 & rng a c= c & c <= b;
    set f = Liouville_seq (a,b);
A2: for i be Nat holds (c-1)/(b to_power (i!)) = ((c-1)(#)(powerfact b)).i
    proof
      let i be Nat;
      ((c - 1) (#) (powerfact b)).i = (c - 1) * ((powerfact b).i) by VALUED_1:6
        .= (c - 1) * (1 / (b to_power (i!))) by DefPower
        .= (c - 1) / (b to_power (i!)) by XCMPLX_1:99;
      hence thesis;
    end;
    let i be Nat;
A4: b >= 1 by A1,XXREAL_0:2;
    reconsider b1 = b - 1 as Element of NAT by A4,INT_1:3,XREAL_1:48;
    reconsider c1 = c - 1 as Element of NAT by A1,INT_1:3,XREAL_1:48;
    per cases;
    suppose
A5:   i is zero; then
A6:   f.i = 0 by DefLio;
A7:   b to_power (i!) = b by NEWTON:12,A5;
      c1 >= 0; then
      (c - 1) / b >= 0;
      hence thesis by A2,A6,A7;
    end;
    suppose
A8:   i is non zero; then
      reconsider ii = i as non zero Nat;
A9:   f.i = (a.i) / (b to_power (i!)) by A8,DefLio;
      reconsider ai = a.i as Nat;
      a.i in rng a by NAT_1:51; then
      a.i in c by A1; then
      a.i in Segm c by ORDINAL1:def 17; then
      ai < c1 + 1 by NAT_1:44; then
      ai <= c1 by NAT_1:13; then
      f.i <= (c - 1) / (b to_power (i!)) by A9,XREAL_1:72;
      hence thesis by A2;
    end;
  end;
