
theorem Th29:
  for a be NAT-valued Real_Sequence, b,n be Nat st b > 0 holds
  ALiouville_seq (a,b).n is Integer
  proof
    let a be NAT-valued Real_Sequence,b,n be Nat;
    set LS = Liouville_seq (a,b);
    set BS = BLiouville_seq b;
    set AS = ALiouville_seq (a,b);
    set ff = BS.n (#) (LS |_ Seg n);
    assume
A0: b > 0;
A1: AS.n = BS.n * Sum (LS |_ Seg n) by ALiuDef;
A2: BS.n * Sum (LS |_ Seg n) = Sum ff by SERIES_1:10;
    rng ff c= INT
    proof
      let y be object;
      assume y in rng ff; then
      consider x be object such that
A3:   x in dom ff & y = ff.x by FUNCT_1:def 3;
      reconsider x as Nat by A3;
A4:   y = (BLiouville_seq b).n * (Liouville_seq (a,b) |_ Seg n).x
        by A3,VALUED_1:6;
      per cases;
      suppose
A5:     x in Seg n; then
A6:     1 <= x <= n by FINSEQ_1:1;
        dom LS = NAT by FUNCT_2:def 1; then
        x in dom (LS | Seg n) by A5,RELAT_1:62; then
A8:     (Liouville_seq (a,b) |_ Seg n).x =
            ((Liouville_seq (a,b) | Seg n)).x by FUNCT_4:13
          .= (Liouville_seq (a,b).x) by FUNCT_1:49,A5;
        (Liouville_seq (a,b).x) * (BLiouville_seq b).n is Integer
          by A0,A6,Th28;
        hence thesis by INT_1:def 2,A4,A8;
      end;
      suppose
        not x in Seg n; then
        not x in dom (Liouville_seq (a,b) | Seg n) by RELAT_1:57; then
        (Liouville_seq (a,b) |_ Seg n).x = (NAT --> 0).x by FUNCT_4:11;
        hence thesis by INT_1:def 2,A4;
      end;
    end; then
    reconsider ff as INT-valued Real_Sequence by RELAT_1:def 19;
    set m = n + 1;
    for k be Nat st k >= m holds ff.k = 0
    proof
      let k be Nat;
      assume k >= m; then
      k > n by NAT_1:13; then
      not k in Seg n by FINSEQ_1:1; then
A9:   not k in dom (LS | Seg n) by RELAT_1:57;
A10:   ff.k = BS.n * ((LS |_ Seg n).k) by VALUED_1:6;
      (LS |_ Seg n).k = (NAT --> 0).k by FUNCT_4:11,A9
        .= 0;
      hence thesis by A10;
    end;
    hence thesis by A1,A2,Th16;
  end;
