
theorem NF200:
  for a being non empty FinSequence of REAL,
  f being FinSequence of NAT, j, b being Nat st
  len f + 1 <= len a holds
  b = j implies
  SumBin (a, f ^ <* b *>, {j}) = SumBin (a, f, {j}) + a . (len f + 1)
  proof
    let a be non empty FinSequence of REAL,
    f be FinSequence of NAT, j, b be Nat;

    assume A10: len f + 1 <= len a;

    assume A15: b = j;

    len (f ^ <* b *>) = len f + 1 by FINSEQ_2:16;
    then A20: dom (f ^ <* b *>) c= dom a by A10,FINSEQ_3:30;

    (f ^ <* b *>) " {j} c= dom (f ^ <* b *>) by RELAT_1:132;
    then A30: (f ^ <* b *>) " {j} c= dom a by A20;

    A40: (f ^ <* b *>) " {j} = f " {j} \/ {len f + 1} by A15,NF100;

    A59: for m being Nat st m in f " {j} holds m < len f + 1
    proof
      let m be Nat;

      assume B00: m in f " {j};

      f " {j} c= dom f by RELAT_1:132;
      then m <= len f by B00,FINSEQ_3:25;
      hence m < len f + 1 by NAT_1:13;
    end;

    Seq (a , (f ^ <* b *>) " {j})
     = Seq (a | (f ^ <* b *>) " {j})
    .= Seq (a | f " {j}) ^ <* a . (len f + 1) *> by A30,A40,A59,NF120
    .= Seq (a , f " {j}) ^ <* a . (len f + 1) *>;
    hence thesis by RVSUM_1:74;
  end;
