
theorem Th31:
  for a be NAT-valued Real_Sequence, b be non zero Nat st
  b >= 2 & rng a c= b holds Liouville_seq (a,b) is summable
  proof
    let a be NAT-valued Real_Sequence, b be non zero Nat;
    assume
A1: b >= 2 & rng a c= b;
A2: b > 1 by A1,XXREAL_0:2; then
A3: b - 1 >= 0 by XREAL_1:48;
    set f = Liouville_seq (a,b);
A4: for i be Nat holds (b-1)/(b to_power (i!)) = ((b-1)(#)(powerfact b)).i
    proof
      let i be Nat;
      ((b - 1)(#)(powerfact b)).i
    = (b - 1)*((powerfact b).i) by VALUED_1:6
   .= (b - 1) * (1 / (b to_power (i!))) by DefPower
   .= (b - 1) / (b to_power (i!)) by XCMPLX_1:99;
      hence thesis;
    end;
A5: for i be Nat holds
      f.i >= 0 & f.i <= ((b - 1) (#) (powerfact b)).i
    proof
      let i be Nat;
      reconsider b1 = b - 1 as Element of NAT by INT_1:3,XREAL_1:48,A2;
      per cases;
      suppose
A6:     i is zero; then
A7:     f.i = 0 by DefLio;
A8:     b to_power (i!) = b by NEWTON:12,A6;
        (b - 1) / b >= 0 by A3;
        hence thesis by A7,A4,A8;
      end;
      suppose
A9:     i is non zero; then
        reconsider ii = i as non zero Nat;
A10:    f.i = (a.i)/(b to_power (i!)) by A9,DefLio;
        reconsider ai = a.i as Nat;
        a.i in rng a by NAT_1:51; then
        a.i in b by A1; then
        a.i in Segm b by ORDINAL1:def 17; then
        ai < b1 + 1 by NAT_1:44; then
        ai <= b1 by NAT_1:13; then
        f.i <= (b - 1) / (b to_power (i!)) by A10,XREAL_1:72;
        hence thesis by A4,A10;
      end;
    end;
    powerfact b is summable by A2,Th26; then
    (b - 1) (#) (powerfact b) is summable by SERIES_1:10;
    hence thesis by A5,SERIES_1:20;
  end;
