
theorem Th1:
  for F being sequence of ExtREAL st for n being Element of NAT
  holds F.n = 0. holds SUM(F) = 0.
proof
  let F be sequence of ExtREAL;
  defpred P[Nat] means Ser(F).$1 = 0.;
  assume
A1: for n being Element of NAT holds F.n = 0.;
A2: for t being Nat st P[t] holds P[t+1]
  proof
    let t be Nat;
    assume Ser(F).t = 0.;
    hence Ser(F).(t+1) = 0. + F.(t+1) by SUPINF_2:def 11
      .= F.(t+1) by XXREAL_3:4
      .= 0. by A1;
  end;
A3: Ser(F).0 = F.0 by SUPINF_2:def 11
    .= 0. by A1;
  then
A4: P[0];
A5: for s being Nat holds P[s] from NAT_1:sch 2(A4,A2);
A6: rng Ser(F) = {0.}
  proof
    thus rng Ser(F) c= {0.}
    proof
      let x be object;
      assume x in rng Ser(F);
      then consider s being object such that
A7:   s in dom Ser(F) and
A8:   x = Ser(F).s by FUNCT_1:def 3;
      reconsider s as Element of NAT by A7;
      Ser(F).s = 0. by A5;
      hence thesis by A8,TARSKI:def 1;
    end;
    let x be object;
    assume x in {0.};
    then dom Ser(F) = NAT & x = 0. by FUNCT_2:def 1,TARSKI:def 1;
    hence thesis by A3,FUNCT_1:def 3;
  end;
  sup {0.} = 0. by XXREAL_2:11;
  hence thesis by A6;
end;
