reserve X for set;

theorem
  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;
  then for n being Element of NAT holds Ser(F2).n <= Ser(F1).n;
  then
A2: SUM(F2) <= SUM(F1) by Th1;
  for n being Element of NAT holds Ser(F1).n <= Ser(F2).n by A1;
  then SUM(F1) <= SUM(F2) by Th1;
  hence thesis by A2,XXREAL_0:1;
end;
