reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem
  for f,g being summable Real_Sequence st for n being Nat
  holds f.n <= g.n holds Sum f <= Sum g
proof
  let f,g be summable Real_Sequence;
A1: Sum f = lim Partial_Sums f & Sum g = lim Partial_Sums g by SERIES_1:def 3;
  assume for n being Nat holds f.n <= g.n;
  then
A2: for n being Nat holds (Partial_Sums f).n <= (Partial_Sums g).
  n by SERIES_1:14;
  Partial_Sums f is convergent & Partial_Sums g is convergent by SERIES_1:def 2
;
  hence thesis by A1,A2,SEQ_2:18;
end;
