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 Th21:
  for P, A, U, u st u in Polish-expression-layer(P, A, U)
      ex p, q st p in P & u = p^q
  proof
  let P, A, U, u;
  assume u in Polish-expression-layer(P, A, U);
  then consider p, q, n such that
  A1: u = p^q & p in P and
      n = A.p & q in U^^n by Def6;
  thus thesis by A1;
  end;
