
theorem
  1/10 < Liouville_constant <= 10/9 - 1/10
  proof
    set x = Liouville_constant;
    set a = seq_const 1;
    set b = 10;
    set c = 2;
A0: rng a = {1} by FUNCOP_1:8;
A1: 1 in 10 by ENUMSET1:def 8,CARD_1:58;
A2: 10 is non trivial by NAT_2:28;
    reconsider n = 1 as non zero Nat;
    set f = Liouville_seq (a,b);
    set pb = powerfact b;
A3: f is summable by Th31,A0,A1,ZFMISC_1:31;
    for n be Nat holds 0 <= f.n by Th33,A2; then
A4: f.1 <= Sum f by RSSPACE2:3,A3;
A5: f.1 = 10 to_power -1 by NEWTON:13,Th39
       .= (1/10) to_power 1 by POWER:32
       .= 1/10;
    set s1 = f ^\ 2;
    set s2 = pb ^\ 2;
A6: (powerfact b).0 = 1/(b to_power(0!)) by DefPower  .= 1/b by NEWTON:12;
A7: (powerfact b).1 = 1/(b to_power (1!)) by DefPower  .= 1/b by NEWTON:13;
A8: (Partial_Sums pb).(0+1) = (Partial_Sums pb).0 + pb.(0+1) by SERIES_1:def 1
                           .= pb.0 + pb.(0 + 1) by SERIES_1:def 1;
    Sum pb = (Partial_Sums pb).1 + Sum (pb^\(1+1)) by Th26,SERIES_1:15
          .= 2/b + Sum s2 by A6,A7,A8; then
A9: Sum s2 = Sum pb - 2 / b;
    Sum pb <= b/(b-1) by Th26; then
A10: Sum s2 <= b/(b-1) - 2/b by A9,XREAL_1:9;
A11: s2 is summable by Th26,SERIES_1:12;
    for n be Nat holds 0 <= s1.n & s1.n <= s2.n
    proof
      let n be Nat;
A12:  s1.n = f.(n + 2) by NAT_1:def 3;
      f.(n+2) <= ((c-1)(#)(powerfact b)).(n+2)
        by Th34,A0,ZFMISC_1:7,CARD_1:50; then
      f.(n+2) <= (c-1)*((powerfact b).(n+2)) by VALUED_1:6;
      hence thesis by Th33,A2,A12,NAT_1:def 3;
    end; then
A13: Sum s1 <= Sum s2 by SERIES_1:20,A11;
A14: Sum f = (Partial_Sums f).1 + Sum (f^\(1+1)) by A3,SERIES_1:15
         .= ((Partial_Sums f).0 + f.(0+1)) + Sum (f^\(1+1)) by SERIES_1:def 1
         .= (f.0 + f.(0+1)) + Sum (f^\(1+1)) by SERIES_1:def 1
         .= (0 + f.1) + Sum (f^\(1+1)) by DefLio
         .= 1/10 + Sum (f^\ 2) by A5;
    Sum s1 <= 10/(10-1) - 2/10 by A13,A10,XXREAL_0:2; then
    1/10 + Sum (f ^\ 2) <= 1/10 + (10/(10 - 1) - 2/10) by XREAL_1:7;
    hence thesis by A4,A5,A14,XXREAL_0:1;
  end;
