reserve X for set;

theorem Th1:
  for F1,F2 being sequence of ExtREAL st (for n being Element
  of NAT holds Ser(F1).n <= Ser(F2).n) holds SUM(F1) <= SUM(F2)
proof
  let F1,F2 be sequence of ExtREAL;
  assume
A1: for n being Element of NAT holds Ser(F1).n <= Ser(F2).n;
A2: for x being ExtReal st x in rng Ser(F1) holds ex y being
  ExtReal st y in rng Ser(F2) & x <= y
  proof
    let x be ExtReal;
A3: dom Ser(F1) = NAT by FUNCT_2:def 1;
    assume x in rng Ser(F1);
    then consider n being object such that
A4: n in NAT and
A5: x = Ser(F1).n by A3,FUNCT_1:def 3;
    reconsider n as Element of NAT by A4;
    reconsider y = Ser(F2).n as R_eal;
    take y;
    dom Ser(F2) = NAT by FUNCT_2:def 1;
    hence thesis by A1,A5,FUNCT_1:def 3;
  end;
  SUM(F1) = sup(rng Ser F1) & SUM(F2) = sup(rng Ser F2)
    by SUPINF_2:def 13;
  hence thesis by A2,XXREAL_2:63;
end;
