reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;

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;
