
theorem Th66:
  for F be FinSequence of ExtREAL holds
     (F is without-infty implies Sum F <> -infty)
   & (F is without+infty implies Sum F <> +infty)
proof
    let F be FinSequence of ExtREAL;
    hereby assume F is without-infty; then
A1:  not -infty in rng F by MESFUNC5:def 3;
     consider S be sequence of ExtREAL such that
A2:   Sum F = S.(len F) & S.0 = 0 &
      for n be Nat st n < len F holds S.(n+1) = S.n + F.(n+1)
        by EXTREAL1:def 2;
     defpred P[Nat] means $1 <= len F implies S.$1 <> -infty;
A3:  P[0] by A2;
A4:  for n be Nat st P[n] holds P[n+1]
     proof
      let n be Nat;
      assume A5: P[n];
      assume A6: n+1 <= len F; then
A7:  S.(n+1) = S.n + F.(n+1) by A2,NAT_1:13;
      n+1 in dom F by A6,NAT_1:11,FINSEQ_3:25; then
      F.(n+1) in rng F by FUNCT_1:3;
      hence S.(n+1) <> -infty by A1,A5,NAT_1:13,A6,A7,XXREAL_3:17;
     end;
     for n be Nat holds P[n] from NAT_1:sch 2(A3,A4);
     hence Sum F <> -infty by A2;
    end;
    assume F is without+infty; then
A8: not +infty in rng F by MESFUNC5:def 4;
    consider S be sequence of ExtREAL such that
A9:  Sum F = S.(len F) & S.0 = 0 &
     for n be Nat st n < len F holds S.(n+1) = S.n + F.(n+1)
       by EXTREAL1:def 2;
    defpred P[Nat] means $1 <= len F implies S.$1 <> +infty;
A10: P[0] by A9;
A11: for n be Nat st P[n] holds P[n+1]
    proof
     let n be Nat;
     assume A12: P[n];
     assume A13: n+1 <= len F; then
A14: S.(n+1) = S.n + F.(n+1) by A9,NAT_1:13;
     n+1 in dom F by A13,NAT_1:11,FINSEQ_3:25; then
     F.(n+1) in rng F by FUNCT_1:3;
     hence S.(n+1) <> +infty by A8,A12,NAT_1:13,A13,A14,XXREAL_3:16;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A10,A11);
    hence Sum F <> +infty by A9;
end;
