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 Th34:
  for P, A, Q st Q is A-closed holds Polish-atoms(P, A) c= Q
proof
  let P, A, Q;
  assume A0: Q is A-closed;
  let a;
  assume A1: a in Polish-atoms(P, A);
  then reconsider a as FinSequence;
  dom A = P by FUNCT_2:def 1;
  then A3: a in dom A & A.a = 0 by A1, Def7;
  {} in Q^^0 by Th4;
  then a^{} in Q by A0, A3;
  hence thesis by FINSEQ_1:34;
end;
