
theorem Th8:
  for F,G being FinSequence of ExtREAL st not -infty in rng F & not
  -infty in rng G holds Sum(F^G) = Sum(F)+Sum(G)
proof
  let F,G be FinSequence of ExtREAL;
  defpred P[FinSequence of ExtREAL] means not -infty in rng $1 implies Sum(F^
  $1) = Sum(F)+Sum($1);
  assume
A1: not -infty in rng F;
A2: for H being FinSequence of ExtREAL for x being Element of ExtREAL st P[H
  ] holds P[H^<*x*>]
  proof
    let H be FinSequence of ExtREAL;
    let x be Element of ExtREAL;
    assume
A3: P[H];
    thus P[H^<*x*>]
    proof
      assume not -infty in rng(H^<*x*>);
      then
A4:   not -infty in rng H \/ rng <*x*> by FINSEQ_1:31;
      then not -infty in rng H by XBOOLE_0:def 3;
      then
A5:   Sum H <> -infty by Lm2;
      not -infty in rng <*x*> by A4,XBOOLE_0:def 3;
      then not -infty in {x} by FINSEQ_1:38;
      then
A6:   x <> -infty by TARSKI:def 1;
A7:   Sum F <> -infty by A1,Lm2;
      F^(H^<*x*>) = F^H^<*x*> by FINSEQ_1:32;
      hence Sum(F^(H^<*x*>)) = Sum F + Sum H + x by A3,A4,Lm1,XBOOLE_0:def 3
        .= Sum F + (Sum H + x) by A6,A7,A5,XXREAL_3:29
        .= Sum F + Sum(H^<*x*>) by Lm1;
    end;
  end;
  assume
A8: not -infty in rng G;
A9: P[<*> ExtREAL]
  proof
    set H = <*> ExtREAL;
    assume not -infty in rng H;
    thus Sum(F^H) = Sum F by FINSEQ_1:34
      .= Sum F + Sum H by Th5,XXREAL_3:4;
  end;
  for H being FinSequence of ExtREAL holds P[H] from FINSEQ_2:sch 2 (A9,
  A2);
  hence thesis by A8;
end;
