The forest contains a Lark L.
With no other information, we can prove that at least one bird in the forest must be egocentric. How?
This one's complicated! Definitely needed to skip to Smullyan's explanation.
From #25, we know that every bird is fond of at least one bird. Suppose that L(L) is fond of y.
Therefore, L(L)(y) => y, but also (from the Lark's definition) L(L)(y) => L(y(y)). Therefore, L(y(y)) === y.
Applying each side to y again, that means that L(y(y))(y) === y(y)—but also, from the Lark's definition, L(y(y))(y) => y(y)(y(y)). Since y(y) === y(y)(y(y)), y(y) is egocentric!