
theorem NF280:
  for a being non empty positive FinSequence of REAL,
  f being FinSequence of NAT, s being set holds
  0 <= SumBin (a, f, s)
  proof
    let a be non empty positive FinSequence of REAL,
        f be FinSequence of NAT, s be set;

    reconsider seqaxs = Seq (a, f " s) as real-valued FinSequence;

    for i being Nat st i in dom seqaxs holds 0 <= seqaxs . i
    proof
      let i be Nat;

      assume i in dom seqaxs; then

      L300: seqaxs . i in rng seqaxs by FUNCT_1:3;

      L400: rng seqaxs = rng (Seq (a | f " s))
      .= rng (a | f " s) by FINSEQ_1:101;

      rng (a | f " s) c= rng a by RELAT_1:70;
      hence thesis by L300,L400,PARTFUN3:def 1;
    end;
    hence 0 <= SumBin (a, f, s) by RVSUM_1:84;
  end;
