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 Th28:
  for P, A, a st a in Polish-expression-set(P, A)
      ex n st a in Polish-expression-hierarchy(P, A, n+1)
proof
  let P, A, a;
  assume A1: a in Polish-expression-set(P, A);
  set Y = the set of all Polish-expression-hierarchy(P, A, n) where
    n is Nat;
  consider X such that A2: a in X and A3: X in Y by A1, TARSKI:def 4;
  consider n such that A4: X = Polish-expression-hierarchy(P, A, n) by A3;
  Polish-expression-hierarchy(P, A, n)
      c= Polish-expression-hierarchy(P, A, n+1) by Th24;
  hence thesis by A2, A4;
end;
