theorem
  for X for B being disjoint_valued FinSequence of bool X for a, n
      st a in B.n holds (arity-from-list B).a = n
proof
  let X;
  let B be disjoint_valued FinSequence of bool X;
  let a, n;
  set F = arity-from-list B;
  assume A2: a in B.n;
  then n in dom B by Th29;
  then A4: B.n in rng B by FUNCT_1:def 3;
  rng B c= bool X by FINSEQ_1:def 4;
  then a in B.(F.a) by A2, A4, Def14;
  hence thesis by A2, Th29;
end;
