
theorem
  for a be NAT-valued Real_Sequence, b, c be Nat
  st b >= 2 & c >= 1 & rng a c= c & c <= b holds
  Sum Liouville_seq (a,b) <= Sum ((c - 1) (#) (powerfact b))
  proof
    let a be NAT-valued Real_Sequence, b, c be Nat;
    assume
A0: b >= 2 & c >= 1 & rng a c= c & c <= b; then
    b >= 1 + 1; then
    b > 1 by NAT_1:13; then
    powerfact b is summable by Th26; then
A1: ((c - 1) (#) (powerfact b)) is summable by SERIES_1:10;
    b is 2_or_greater by A0,EC_PF_2:def 1; then
A2: for i be Nat holds 0 <= Liouville_seq (a,b).i by Th33;
    for i be Nat holds Liouville_seq (a,b).i <= ((c-1)(#)(powerfact b)).i
    by Th34,A0;
    hence thesis by SERIES_1:20,A1,A2;
  end;
