
theorem Th48:
  for F being sequence of ExtREAL st (for n being Element of
NAT holds F.n in REAL) holds for n being Nat holds Ser(F).n in REAL
proof
  let F be sequence of ExtREAL;
  defpred P[Nat] means Ser(F).$1 in REAL;
  assume
A1: for n being Element of NAT holds F.n in REAL;
A2: for s being Nat st P[s] holds P[s+1]
  proof
    let s be Nat;
    reconsider b = F.(s+1) as Element of REAL by A1;
    reconsider y = Ser(F).s as R_eal;
    assume Ser(F).s in REAL;
    then reconsider a = y as Element of REAL;
A3: y + F.(s+1) = a + b by XXREAL_3:def 2;
    Ser(F).(s+1) = y + F.(s+1) by Def11;
    hence thesis by A3;
  end;
  Ser(F).0 = F.0 by Def11; then
A4: P[0] by A1;
  thus for s being Nat holds P[s] from NAT_1:sch 2(A4,A2);
end;
