Complete Codes and Kraft Inequality

Question

Li and Vitanyi define a complete code as a uniquely decodable code to which no codeword can be added while keeping it uniquely decodable. They claim that this is easily seen to be equivalent to equality holding in the Kraft inequality. I can see that equality in the Kraft inequality is clearly a necessary condition, and the other direction is obvious to me for prefix codes, but otherwise it does not seem so easy to me.