The two iterated integrals are therefore equal. On the other hand, since is continuous, the second iterated integral can be performed by first integrating over and then afterwards over . But then the iterated integral of on must vanish. However, if the iterated integral of a continuous function function vanishes for all rectangles, then must be identically zero; for otherwise or would be strictly positive at some point and therefore by continuity on a rectangle, which is not possible. Hence must vanish identically, so that everywhere.