reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem Th56:
  for f being Function of X, X st f is one-to-one & rng f = X
  holds f is Permutation of X
proof
  let f be Function of X, X;
  assume that
A1: f is one-to-one and
A2: rng f = X;
  f is onto by A2;
  hence thesis by A1;
end;
