
theorem Th47:
  for F being sequence of ExtREAL st F is nonnegative holds
for n being Nat st (for r being Element of NAT st n <= r holds F.r = 0.) holds
  SUM(F) = Ser(F).n
proof
  let F be sequence of ExtREAL;
  assume
A1: F is nonnegative;
  let n be Nat;
  reconsider n9=n as Element of NAT by ORDINAL1:def 12;
  assume
A2: for r being Element of NAT st n <= r holds F.r = 0.;
A3: for r being Element of NAT st n <= r holds Ser(F).r <= Ser(F).n
  proof
    defpred P[Nat] means Ser(F).(n + $1) <= Ser(F).n;
    let r be Element of NAT;
    assume n <= r;
    then
A4: ex m being Nat st r = n + m by NAT_1:10;
A5: for s being Nat st P[s] holds P[s+1]
    proof
      let s be Nat;
      reconsider s9=s as Element of NAT by ORDINAL1:def 12;
      reconsider y = Ser(F).(n + s) as R_eal;
      n9 + (s9 + 1) = (n9 + s9) + 1; then
A6:   Ser(F).(n9 + (s9 + 1)) = y + F.(n9 + (s9 + 1)) by Def11;
      n + 0 <= n + (s+1) by XREAL_1:7; then
A7:   F.(n9 + (s9+1)) = 0. by A2;
      assume Ser(F).(n + s) <= Ser(F).n;
      hence thesis by A6,A7,XXREAL_3:4;
    end;
A8: P[0];
    for m being Nat holds P[m] from NAT_1:sch 2(A8,A5);
    hence thesis by A4;
  end;
A9: for r being Element of NAT st r <= n holds Ser(F).r <= Ser(F).n
  proof
    let r be Element of NAT;
    assume r <= n;
    then consider m being Nat such that
A10: n = r + m by NAT_1:10;
    reconsider m as Element of NAT by ORDINAL1:def 12;
    thus thesis by A1,Th40,A10;
  end;
  for y being ExtReal st y in rng Ser(F) holds y <= Ser(F).n
  proof
    let y be ExtReal;
A11: dom Ser(F) = NAT by FUNCT_2:def 1;
    assume y in rng Ser(F);
    then consider m being object such that
A12: m in NAT and
A13: y = Ser(F).m by A11,FUNCT_1:def 3;
    reconsider m as Element of NAT by A12;
    m <= n or n <= m;
    hence thesis by A3,A9,A13;
  end;
  then
A14: Ser(F).n is UpperBound of rng Ser(F) by XXREAL_2:def 1;
  Ser(F).n9 in rng Ser(F) by FUNCT_2:4;
  hence thesis by A14,XXREAL_2:55;
end;
