reserve a, b, p, q for Real;

theorem
  for p be Real st 0 < p for a,ap be Real_Sequence st a is summable & (
for n be Nat holds 0 <=a.n ) & (for n be Nat holds ap.n=(
  Partial_Sums(a).n) to_power p) holds ap is convergent & lim ap = Sum(a)
to_power p & ap is non-decreasing & for n be Nat holds ap.n <= Sum(a
  ) to_power p
proof
  let p be Real such that
A1: 0 < p;
  let a,ap be Real_Sequence such that
A2: a is summable and
A3: for n be Nat holds 0 <=a.n and
A4: for n be Nat holds ap.n=(Partial_Sums(a).n) to_power p;
A5: Partial_Sums(a) is convergent & for n be Nat holds 0 <=
  Partial_Sums(a).n by A2,A3,Lm2,SERIES_1:def 2;
  then lim ap = (lim Partial_Sums(a)) to_power p by A1,A4,Th10;
  hence
A6: ap is convergent & lim ap = (Sum(a)) to_power p by A1,A4,A5,Th10,
SERIES_1:def 3;
A7: Partial_Sums(a) is non-decreasing by A3,SERIES_1:16;
  now
    let n, m be Nat;
    assume n <=m;
    then
A8: Partial_Sums(a).n <= Partial_Sums(a).m by A7,SEQM_3:6;
A9: ap.n = (Partial_Sums(a).n) to_power p & ap.m = (Partial_Sums(a).m)
    to_power p by A4;
    0 <= Partial_Sums(a).n by A3,Lm2;
    hence ap.n <= ap.m by A1,A9,A8,Th3;
  end;
  hence
A10: ap is non-decreasing by SEQM_3:6;
  thus thesis by A6,A10,SEQ_4:37;
end;
