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 Th39:
  for P, A, n, p, q st p in P & n = A.p & q in Polish-expression-set(P, A)^^n
      holds Polish-operation(P, A, n, p).q in Polish-expression-set(P, A)
proof
  let P, A, n, p, q;
  set U = Polish-expression-set(P, A);
  assume A1: p in P & n = A.p & q in U^^n;
  A2: U is A-closed;
  A3: dom A = P by FUNCT_2:def 1;
  dom Polish-operation(P, A, n, p) = U^^n by FUNCT_2:def 1;
  then Polish-operation(P, A, n, p).q = p^q by Def12, A1;
  hence thesis by A1,A2,A3;
end;
