
theorem Th17:
for F be FinSequence of ExtREAL st
 for i be Nat st i in dom F holds F.i <> -infty
holds Sum F <> -infty
proof
   let F be FinSequence of ExtREAL;
   assume A1: for i be Nat st i in dom F holds F.i <> -infty;
   consider f be Function of NAT,ExtREAL such that
A2: Sum F = f.(len F) & f.0 = 0 &
    for i be Nat st i < len F holds f.(i+1) = f.i + F.(i+1)
      by EXTREAL1:def 2;
   defpred P[Nat] means $1 <= len F implies f.$1 <> -infty;
A4:P[0] by A2;
A5:for j be Nat st P[j] holds P[j+1]
   proof
    let j be Nat;
    assume A6: P[j];
    now assume B2: j+1 <= len F; then
A8:  f.(j+1) = f.j + F.(j+1) by A2,NAT_1:13;
     1 <= j+1 by NAT_1:11; then
     F.(j+1) <> -infty by A1,B2,FINSEQ_3:25;
     hence f.(j+1) <> -infty by A8,A6,B2,NAT_1:13,XXREAL_3:17;
    end;
    hence P[j+1];
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A4,A5);
   hence Sum F <> -infty by A2;
end;
