
theorem Th17:
  for p be FinSequence of ExtREAL st not -infty in rng p & +infty in rng p
  holds Sum(p) = +infty
proof
  let p be FinSequence of ExtREAL;
  assume
A1: not -infty in rng p;
   assume +infty in rng p;
   then ex n be object st n in dom p & p.n=+infty by FUNCT_1:def 3;
   then consider m be Nat such that
A2: m in dom p and
A3: p.m = +infty;
  m in Seg len p by A2,FINSEQ_1:def 3;
  then
A4: len p >= m by FINSEQ_1:1;
  consider f be sequence of ExtREAL such that
A5: Sum(p) = f.(len p) and
A6: f.0 = 0. and
A7: for i be Nat st i < len p holds f.(i+1)=f.i + p.(i+1)
    by EXTREAL1:def 2;
A8: for n be Nat st n in dom p holds -infty < p.n
    proof
      let n be Nat;
      assume n in dom p;
      then p.n in rng p by FUNCT_1:def 3;
      hence thesis by A1,XXREAL_0:6;
    end;
  defpred P[Nat] means $1 <= len p implies
  ($1 < m implies -infty < f.$1) & ($1 >= m implies f.$1 = +infty);
A9: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume
A10: P[i];
    assume
A11: i+1 <= len p;
    reconsider i as Element of NAT by ORDINAL1:def 12;
A12: i < len p by A11,NAT_1:13;
    now
      per cases;
      case i+1 < m;
        1 <= i+1 by NAT_1:11;
        then i+1 in Seg len p by A11,FINSEQ_1:1;
        then i+1 in dom p by FINSEQ_1:def 3;
        then
A13:    -infty < p.(i+1) by A8;
A14:    -infty + -infty = -infty by XXREAL_3:def 2;
        -infty + -infty < f.i + p.(i+1) by A10,A11,A13,NAT_1:13,XXREAL_3:64;
        hence -infty < f.(i+1) by A7,A12,A14;
      end;
      case
A15:    i+1 >= m;
        now
          per cases;
          case
A16:        i+1 = m;
            f.(i+1) = f.i + p.(i+1) by A7,A12;
            hence f.(i+1) = +infty by A3,A10,A11,A16,NAT_1:13,XXREAL_3:def 2;
          end;
          case
A17:        i+1 <> m;
            i < len p by A11,NAT_1:13;
            then
A18:        f.(i+1) = f.i + p.(i+1) by A7;
            1 <= i+1 by NAT_1:11;
            then i+1 in Seg len p by A11,FINSEQ_1:1;
            then i+1 in dom p by FINSEQ_1:def 3;
            then
A19:        p.(i+1) <> -infty by A8;
            i+1 > m by A15,A17,XXREAL_0:1;
            hence f.(i+1) = +infty by A10,A11,A19,A18,NAT_1:13,XXREAL_3:def 2;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A20: P[0] by A2,A6,FINSEQ_3:25;
  for i be Nat holds P[i] from NAT_1:sch 2(A20,A9);
  hence thesis by A5,A4;
end;
