
theorem Th41:
  for F1,F2 being sequence of ExtREAL st
    (for n being Element of NAT holds F1.n <= F2.n) holds
    for n being Element of NAT holds Ser(F1).n <= Ser(F2).n
proof
  let F1,F2 be sequence of ExtREAL;
  defpred P[Nat] means Ser(F1).$1 <= Ser(F2).$1;
  assume
A1: for n being Element of NAT holds F1.n <= F2.n;
  let n be Element of NAT;
A3: Ser(F2).0 = F2.0 by Def11;
A5: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A6: Ser(F1).k <= Ser(F2).k;
A7: F1.(k+1) <= F2.(k+1) by A1;
A8: Ser(F2).(k+1) = Ser(F2).k + F2.(k+1) by Def11;
    Ser(F1).(k+1) = Ser(F1).k + F1.(k+1) by Def11;
    hence thesis by A6,A8,A7,XXREAL_3:36;
  end;
  Ser(F1).0 = F1.0 by Def11;
  then
A9: P[0] by A1,A3;
  for n being Nat holds P[n] from NAT_1:sch 2(A9,A5);
  hence thesis;
end;
