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 Th32:
  for P, A, r st r in Polish-expression-set(P, A)
      ex n, p, q
          st p in P & n = A.p & q in Polish-expression-set(P, A)^^n & r = p^q
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 p, q, n such that
  A2: r = p^q and
  A3: p in P and
  A4: n = A.p and
  A5: q in U^^n by Def6;
  take n, p, q;
  U^^n c= Polish-expression-set(P, A)^^n by Th26, Th17;
  hence thesis by A2, A3, A4, A5;
end;
