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 Th18:
  for P, A, U, n, p, q st p in P & n = A.p & q in U^^n holds
      p^q in Polish-expression-layer(P, A, U)
proof
  let P, A, U, n, p, q;
  set r = p^q;
  assume that
  A1: p in P and
  A2: n = A.p and
  A3: q in U^^n;
  A4: q in P* by A3, Th14, TARSKI:def 3;
  p in P* by A1, Th9, TARSKI:def 3;
  then r in P* by A4, Th12;
  hence thesis by A1, A2, A3, Def6;
end;
