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 Th19:
  for P, A, U holds Polish-atoms(P, A) c= Polish-expression-layer(P, A, U)
proof
  let P, A, U;
  set X = Polish-atoms(P, A);
  set Y = Polish-expression-layer(P, A, U);
  let a;
  assume A1: a in X;
  then reconsider p = a as FinSequence;
  set q = <*>P;
  A3: q in U^^0 by Th4;
  p in P & A.p = 0 by A1, Def7; then
  p^q in Y by A3, Th18;
  hence thesis by FINSEQ_1:34;
end;
