
theorem Th5:
  for F,G being sequence of ExtREAL holds ( for n being Element
of NAT holds F.n <= G.n) implies for n being Element of NAT holds (Ser(F)).n <=
  SUM(G)
proof
  let F,G be sequence of ExtREAL;
  assume
A1: for n being Element of NAT holds F.n <= G.n;
  let n be Element of NAT;
  set y = Ser(G).n;
  dom Ser(G) = NAT by FUNCT_2:def 1;
  then SUM(G) = sup(rng Ser(G)) & y in rng Ser(G) by FUNCT_1:def 3;
  hence thesis by A1,SUPINF_2:42,XXREAL_2:61;
end;
