. Since
was any one-one correlation of all members with some sub-classes, it follows that there is no correlation of all members with all sub-classes. It does not matter to the proof if
has no members: all that happens in that case is that the sub-class which is shown to be omitted is the null-class. Hence in any case the number of sub-classes is not equal to the number of members, and therefore, by what was said earlier, it is greater. Combining this with the proposition that, if
is the number of members,
is the number of sub-classes, we have the theorem that