
theorem Th35:
  for a be NAT-valued Real_Sequence, b, c, n be Nat st
  b >= 2 & c >= 1 & rng a c= c & c <= b holds
  Sum (Liouville_seq (a,b)^\(n+1)) <= Sum ((c-1)(#)((powerfact b)^\(n+1)))
  proof
    let a be NAT-valued Real_Sequence, b, c, n be Nat;
    set g = (c-1)(#)((powerfact b)^\(n+1));
    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)^\(n+1) is summable by Th26,SERIES_1:12; then
A1: (c-1)(#)((powerfact b)^\(n+1)) is summable by SERIES_1:10;
    set f = (Liouville_seq (a,b)^\(n+1));
A2: b is 2_or_greater by A0,EC_PF_2:def 1;
A3: for i be Nat holds 0 <= f.i
    proof
      let i be Nat;
      dom f = NAT by FUNCT_2:def 1; then
      f.i in rng f by FUNCT_1:3,ORDINAL1:def 12;
      hence thesis by A2,PARTFUN3:def 4;
    end;
    for i be Nat holds f.i <= g.i
    proof
      let i be Nat;
A4:   f.i = Liouville_seq (a,b).(n+1+i) by NAT_1:def 3;
      g.i = (c-1)*(((powerfact b)^\(n+1)).i) by SEQ_1:9
         .= (c-1)*((powerfact b).(n+1+i)) by NAT_1:def 3
         .= ((c-1)(#)(powerfact b)).(n+1+i) by SEQ_1:9;
      hence thesis by A4,Th34,A0;
    end;
    hence thesis by SERIES_1:20,A1,A3;
  end;
