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 Th23:
  for P, A, n holds Polish-expression-hierarchy(P, A, n+1)
      = Polish-expression-layer(P, A, Polish-expression-hierarchy(P, A, n))
proof
  let P, A, n;
  consider U such that
  A1: U = Polish-expression-hierarchy(P, A, n) and
  A2: Polish-expression-hierarchy(P, A, n+1)
          = Polish-expression-layer(P, A, U) by Def9;
  thus thesis by A1, A2;
end;
