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 Th25:
  for P, A, n, m holds
    Polish-expression-hierarchy(P, A, n)
        c= Polish-expression-hierarchy(P, A, n+m)
proof
  let P, A, n, m;
  defpred S[ Nat] means
    Polish-expression-hierarchy(P, A, n)
        c= Polish-expression-hierarchy(P, A, n+$1);
  A1: S[ 0 ];
  A2: for k holds S[ k ] implies S[ k + 1 ]
  proof
    let k;
    assume A3: S[ k ];
    Polish-expression-hierarchy(P, A, n+k)
        c= Polish-expression-hierarchy(P, A, n+k+1) by Th24;
    hence thesis by A3, XBOOLE_1:1;
  end;
  for k holds S[ k ] from NAT_1:sch 2(A1, A2);
  hence thesis;
end;
