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 P, A, r st r in Polish-expression-set(P, A)
      ex n, t, q st t in P & n = A.t & r = Polish-operation(P, A, n, t).q
          & q in Polish-expression-set(P,A)^^n
proof
  let P, A, r;
  assume r in Polish-expression-set(P, A);
  then consider m such that A1: r in Polish-expression-hierarchy(P, A, m+1)
      by Th28;
  set U = Polish-expression-hierarchy(P, A, m);
  r in Polish-expression-layer(P, A, U) by A1, Th23;
  then consider t, q, n such that
  A4: r = t^q and
  A5: t in P and
  A6: n = A.t and
  A7: q in U^^n by Def6;
  take n, t, q;
  A10: U^^n c= Polish-expression-set(P,A)^^n by Th17, Th26;
  then U^^n c= dom Polish-operation(P, A, n, t) by FUNCT_2:def 1;
  hence thesis by A4, A5, A6, A7, A10, Def12;
end;
