
theorem NF330:
  for a being non empty at_most_one FinSequence of REAL holds
  [/ Sum a \] <= Opt a
  proof
    let a be non empty at_most_one FinSequence of REAL;

    consider g1 being non empty FinSequence of NAT such that
    L00: dom g1 = dom a and
    L10: (for j being Nat st j in rng g1 holds SumBin (a, g1, {j}) <= 1) and
    L20: Opt a = card rng g1 and
    (for f being non empty FinSequence of NAT st
    dom f = dom a &
    (for j being Nat st j in rng f holds SumBin (a, f, {j}) <= 1)
    holds Opt a <= card rng f) by defOpt;

    thus [/ Sum a \] <= Opt a by L00,L10,NF320,L20;
  end;
