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 Th20:
  for P, A, U, V st U c= V holds
      Polish-expression-layer(P, A, U) c= Polish-expression-layer(P, A, V)
proof
  let P, A, U, V;
  assume A1: U c= V;
    let a;
    assume A2: a in Polish-expression-layer(P, A, U);
    then consider p, q, n such that
A4: a = p^q & p in P & n = A.p & q in U^^n by Def6;
    U^^n c= V^^n by A1, Th17;
    hence thesis by A2, A4, Def6;
end;
