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 Th37:
  for P, A, Q st Q is A-closed holds Polish-expression-set(P, A) c= Q
proof
  let P, A, Q;
  assume A1: Q is A-closed;
  let a;
  assume a in Polish-expression-set(P, A);
  then consider n such that A2: a in Polish-expression-hierarchy(P, A, n+1)
      by Th28;
  thus thesis by A1, A2, Th35, TARSKI:def 3;
end;
