
theorem Th28:
  for a be NAT-valued Real_Sequence,b,n,k be Nat st b > 0 & k <= n holds
  (Liouville_seq (a,b).k) * (BLiouville_seq b).n is Integer
  proof
    let a be NAT-valued Real_Sequence, b,n,k be Nat;
    assume
A1: b > 0 & k <= n; then
    0 + k! <= n! by SIN_COS:39; then
A2: n! - k! in NAT by INT_1:3,XREAL_1:19;
    set bk = b to_power (k!), bn = b to_power (n!);
A3: bn / bk = b to_power (n! - k!) by A1,POWER:29;
    per cases;
    suppose k = 0; then
      Liouville_seq (a,b).k = 0 by DefLio;
      hence thesis;
    end;
    suppose
      k is non zero; then
      (Liouville_seq (a,b)).k = (a.k)/(b to_power (k!)) by DefLio; then
      (Liouville_seq (a,b).k) * (BLiouville_seq b).n
    = ((a.k)/(b to_power (k!))) * ((b to_power(n!) /1)) by LiuSeq
   .= ((a.k) * bn)/(bk * 1) by XCMPLX_1:76
   .= (a.k) * (bn/bk) by XCMPLX_1:74;
      hence thesis by A2,A3;
    end;
  end;
