
theorem Th8:
  for F being sequence of ExtREAL st F is nonnegative holds for
  n,k being Nat st n <= k holds Ser(F).n <= Ser(F).k
proof
  let F be sequence of ExtREAL;
  assume
A1: F is nonnegative;
  let n,k be Nat;
  defpred P[Nat] means Ser(F).n <= Ser(F).(n+$1);
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: Ser(F).n <= Ser(F).(n+k);
    Ser(F).(n+k) <= Ser(F).((n+k)+1) by A1,SUPINF_2:40;
    hence thesis by A3,XXREAL_0:2;
  end;
  assume n <= k;
  then consider s being Nat such that
A4: k = (n qua Complex) + s by NAT_1:10;
A5: k = n + s by A4;
A6: P[0];
  for s being Nat holds P[s] from NAT_1:sch 2(A6,A2);
  hence thesis by A5;
end;
