
theorem Th40:
  for F being sequence of ExtREAL st F is nonnegative holds
  for n,m being Nat holds Ser(F).n <= Ser(F).(n + m)
proof
  let F be sequence of ExtREAL;
  assume A0:F is nonnegative;
  let n,m be Nat;
  defpred P[Nat] means Ser(F).n <= Ser(F).(n + $1);
A1: for k being Nat st P[k] holds P[k + 1]
  proof
    let k be Nat;
    Ser(F).(n + (k+1)) = Ser(F).((n + k) + 1); then
A2: Ser(F).(n + k) <= Ser(F).(n + (k+1)) by Th39,A0;
    assume Ser(F).n <= Ser(F).(n + k);
    hence thesis by A2,XXREAL_0:2;
  end;
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
