
theorem Th42:
  for F1,F2 being sequence of ExtREAL st
  (for n being Element of NAT holds F1.n <= F2.n) holds SUM(F1) <= SUM(F2)
proof
  let F1,F2 be sequence of ExtREAL;
  assume
A1: for n being Element of NAT holds F1.n <= F2.n;
  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;
A2: dom Ser(F1) = NAT by FUNCT_2:def 1;
    assume x in rng Ser(F1);
    then consider n being object such that
A3: n in NAT and
A4: x = Ser(F1).n by A2,FUNCT_1:def 3;
    reconsider n as Element of NAT by A3;
    reconsider y = Ser(F2).n as R_eal;
    take y;
    dom Ser(F2) = NAT by FUNCT_2:def 1;
    hence thesis by A1,A4,Th41,FUNCT_1:def 3;
  end;
  hence thesis by XXREAL_2:63;
end;
