
theorem NF270:
  for a being non empty FinSequence of REAL,
  f being FinSequence of NAT, s, t being set st s misses t holds
  SumBin (a, f, s \/ t) = SumBin (a, f, s) + SumBin (a, f, t)
  proof
    let a be non empty FinSequence of REAL,
    f be FinSequence of NAT, s, t be set;

    assume
    L08: s misses t;

    set U = f " s, V = f " t;

    reconsider F = a as PartFunc of NAT,REAL;

    A200: dom (F | (U \/ V)) is finite;

    A290: for W be set holds Sum (F, W) = Sum (Seq (a, W))
    proof
      let W be set;
      dom (F | (W /\ W)) is finite;
      then A310: (a | W), FinS (F,W) are_fiberwise_equipotent
      by RFUNCT_3:def 13;

      reconsider fssu = a | W as Subset of a by RELAT_1:59;

      A320: Seq fssu, fssu are_fiberwise_equipotent by DBLSEQ_2:51;
      thus Sum (F, W) = Sum FinS (F,W) by RFUNCT_3:def 14
      .= Sum Seq (a, W) by A310,A320,CLASSES1:76,RFINSEQ:9;
    end;
    A610: Sum (F,U) = SumBin (a, f, s) by A290;
    A620: Sum (F,V) = SumBin (a, f, t) by A290;
    Sum (F,(U \/ V))
     = Sum (Seq (a, (f " s \/ f " t))) by A290
    .= SumBin (a, f, s \/ t) by RELAT_1:140;
    hence SumBin (a, f, s \/ t) = SumBin (a, f, s) + SumBin (a, f, t)
    by A200,L08,FUNCT_1:71,RFUNCT_3:83,A610,A620;
  end;
