
theorem Th32:
  for a be Real_Sequence,n be non zero Nat, b be non zero Nat st b > 1
  holds (ALiouville_seq(a,b)).n /(BLiouville_seq b).n
  = Sum FinSeq (Liouville_seq (a,b),n)
  proof
    let a be Real_Sequence, n, b be non zero Nat;
    assume b > 1; then
A1: (BLiouville_seq b).n <> 0 by Th30;
A2: Liouville_seq (a,b).0 = 0 by DefLio;
    thus (ALiouville_seq (a,b)).n / (BLiouville_seq b).n
       = (BLiouville_seq b).n * Sum (Liouville_seq (a,b) |_ Seg n) /
      (BLiouville_seq b).n by ALiuDef
      .= Sum (Liouville_seq (a,b) |_ Seg n) by XCMPLX_1:89,A1
      .= Sum FinSeq (Liouville_seq (a,b),n) by Th18,A2;
  end;
